The nonstationarity of systematic risk for bonds

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FACULTY UORKIWG PAPERS

College of Commerce and Business Administration

University of Illinois at Urb ana- Champaign

July 20, 1978

THE NONSTATIONAPvITY OF SYSTEIIATIC RISK FOR BONDS

Ali Jahankhani, A'^';istant Professor
George E. Pinches, University of Kansas

/M97

Summary;

Recently a number of researchers have attempted to employ the
market model to estimate systematic risk (i.e., beta) for bonds. In
this study we reviewed theoretical evidence which suggests bond
betas can be expected to be nonstatlonary. This nonstationarity is
a function of the duration of a bond, the standard deviation of the
change in the yield to maturity of a bond relative to the standard
deviation of the return on the market portfolio, and the correlation
betv7een the change in the yield to maturity of a bond and the return
on the market portfolio. Hotjever, all bonds will not necessarily
have nonstatlonary betas in a given time period since it is possible
that these factors may occasionally counteract one another.

Empirical tests indicated that over 80 percent of the bonds ejc»
amined had nonstatlonary betas. The primary factor differentiating
bonds with nonstatlonary betas from those with stationary betas was
the substantially higher relative standard deviation in the change in
the yield to maturity for bonds with nonstatlonary betas. The larger
standard deviation was caused by the higher average coupon rates and
yields to maturity for bonds \rLth nonstatlonary betas. The theoretical
and empirical results of this study indicate bond betas, in general,
tend to be nonstatlonary. Hence, further use of them appears to be
of very questionable value.

THE NONSTATIONARITY OF SYSTEMATIC RISK FOR BONDS

I, INTRODUCTION
In the last decade Increasing use has been made of the Capital Asset
PrlclTig Model (CAFM) developed by Sharpe [31], Llntner [7] and extended
by Bl£.ck [2]. In this model the only relevant risk of an asset Is the
systematic risk which Is measured by the covarlance of the ex-ante return
on th(t asset with the ex-ante return on the market portfolio. Since the
ex-anl.e returns cannot be observed, researchers have used historical data
to estimate the systematic risk. The market model, which has been the
most common method of estimating the relative systematic risk (3.) states
that:

R.^ - a. + e.R ^ + e.^, (1)

it 1 1 mt it'

where R. . and R are returns on 1 asset and the market portfolio

2
respeiitively, and B. is computed as cov(R. ,R )/o (R ). The use of

historical data to estimate B. is justified only if the Joint distribu-
tion of returns on the asset and the market portfolio is stable over time.
Under these conditions 3. will be stationary and hence the market model
will ]>e an appropriate method of estimating fi..

]lecently a number of researchers have applied the market model to
estlxuite systematic risk for bonds. Percival [27] estimated bond betas
and tlien attempted to explain them as a function of the bond character-
istic)}. Friend and Blume [14] and McCallum [21] also estimated bond
systematic risk while Rellly and Joehnk [28] examined the relationship
betwe>3n bond betas and bond ratings. Finally, Warner [34] estimated

3 '-■

C ■.... J<i;j;!"'!

-2-

bond betas and then examined the risk adjusted performance of bonds for
firms in bankruptcy versus the performance of bonds for similar firms
not in bankruptcy.

The Increasing use of the bond betas appears to be without support
since there are theoretical considerations suggesting bond betas are
inherently nonstationary. In the presence of this nonstatlonarlty , bond
betas appear to be poor estimates of systematic (or any other kind of)
risk for bonds. The purposes of this stxidy are threefold: 1) to examine
the theoretical considerations indicating nonstatlonarlty of bond betas;
2) to test empirically whether the betas of individual bonds are stationary
over the 1969-1975 time period; and 3) to explain the observed stationarlty/
nonstationarity in terms of the factors that cause the nonstatlonarlty
In bond beta. In Section II theoretical arguments for the nonstatlonarlty
of bond betas are reviewed, while Section III contains the methodology em-
ployed. The empirical results are presented and discussed in Section IV
and the conclusions are contained in Section V.

II. THEOEIETICAL CONSIDERATIONS
Several recent studies [4,15,18,35] have examined a specific
time-risk relationship using a measure of time known as duration. The
concept of duration was first introduced by Macaulay [19] in his study
of bond yields. Unlike the time to maturity, which looks only at the
last payment, duration gives some weight to the time at which each cash
payment is received. The weight assigned to each period is the present
value of the cash payment for that period divided by the current market
price of the security. For a bond, duration at time t^ is computed as:

-3-

N N

D^ - [ Z (t.-tf.) A 1 / 2 A (2)

where A Is the present value measured at time t^ of cash flows to be

3
received at time t. and N is the number of years to maturity. From (2)

it is apparent that duration is a function of the time to maturity, the
size of interim coupon payments, the yield to maturity and the size of the
principal payment. For a zero-coupon bond, duration is identical to the
time to maturity. The litik between the bond price volatility and dura-
tion was developed by Fisher [13] and extended by Hopewell and Kaufman [15] .
Assuming continous compounding, the percentage change in a bond price is
related to duration by:
dP,

^' - -D,, dy,,, (3>

^it ^' '^^

where dP. and P. are the price change and initial price of bond i at
time t respectively. D. is the duration of the bond at time t and dy.
is the change in the yield to maturity. Equation (3) shows that duration
is a constant of proportionality relating percentage changes in boiui prices
to changes in the yield i^Vj^^ •

Boquist, Racette and Schlarbaum [4] developed a theoretical model
which links the beta of a default free bond to duration:

cov(dy ,R ) o(dy )

^it - '\t i'" ; ° -^it ^^'Ht>\t^ Torr ' <^>

a (R^j.) mt

where a(dy. ) is the standard deviation of dy^ , o(R^^) is the standard
deviation of the return on the market portfolio, and P(*iy4t»^mt^ ^® *^^®
correlation coefficient between changes in the yield to maturity and the
return on the market portfolio. (As argued by Boquist et al., the corre-
lation coefficient is expected to be negative for most bonds.) From

-4-

eqiiation (4) it is apparent that &. is dependent upon the duration of
the bond, the correlation coefficient between changes in the yield to
maturity of the bond and the return on the market, and the standard devia-
tion of the changes in the yield to maturity for the bond relative to the
standard deviation of the return on the market portfolio. Therefore,
depending upon the interaction of changes over time in the following three

factors: 1) U^^; 2) -P(^yit'\t^' ^^ ^^ °^^^±t^^°^\t^ ^^^ ^°^ ^^^^
may be stationary or nonstantionary. As a bond progresses toward
maturity the duration, D. , will shorten which, ceteris paribus, should
cause 3. to decrease. The second factor, -p(dy. ,R ), may also cause
p. to decrease over time. Through the passage of time the maturity of
a bond becomes shorter. In general short-term yields tend to be less cor-
related with the return on the market portfolio than the long-term yields.

2
Therefore as time passes the second factor will cause ^. to decrease.

Finally, the third factor, o(dy, )/o(R ), should cause p. to increase
because, as Malkiel [20] has shown, short-term yields tend to be more
volatile than long-term yields. Unless these factors exactly offset each
other bond betas estimated from historical time series data will be non-
stationary.

III. METHODOLOGY
A. SAMPLE

In order to empirically test for the nonstationarity of bond
systematic risk a homogeneous group of bonds was required. The selection
criteria employed resulted in bonds being selected if they were public
utility or industrial bonds continuously rated (without any change) in
the top four bond rating categories by both Moody's and Standard & Poor's

-5-

between May 31, 1969 and May 31, 1975, were issued between January 1,

1966 and March 1, 1969, had an original maturity of at le^tst 20 years

3

and an original issue size of at least !?10 million. In jiddition, the

bonds could not be subordinated or convertible, nor could they be Issued

4
with warrants attached. In cases where there were more than one bond

per company that met the selection criteria, the most rectsnt issue was

selected. Application of these criteria resulted in 84 bonds being

selected of which 42 were public utility bonds and 42 were industrial

bonds.

B. VARIABLES

1. Holding Period Return

Monthly holding period returns for bonds were comput(id as:

- I + AP^

where I is the periodic interest payment per $100 of par value;
m is the number of holding periods between interest payments (for most
bonds m = 6 months) ; n is the number of periods accrued toward the next
interest payment at the end of period t; and P - is the laarket price
of the bond at the end of period t-1.

Some authors have used different methods to measure the riiturn on bonds*
Yawitz and Marshall [36] used purchase yield as a measure of return on
U.S. Government bonds. They reasoned that it is a better measure of the
expected return because over the life of the bond, price i:hanges must
stim to zero. This argiunent is valid only if the investorii' holding period
is equal to the life of the bond. Yield to maturity has also been used
as a measure of the returns on bonds by Duvall and Cheney [10] . They

Ir

;3i', S;.:

-6-

argued that yield to maturity is a more reliable estimate of the expected
return than the ex-post measure as formulated in equation (5). Reilly and
Joehnk [28] employed the percentage change in the yield to maturity as a
measure of the bond return. Again these authors are implicitly assuming
that investors have a holding period equal to the life of the bond, the
bond Is default free, and investors can reinvest the intermediate interest
payments at a rate equal to the yield to maturity. Because of the above
mentioned problems with these return measures, we prefer to use equation
(5) to measure the holding period return.

2. Market Portfolio

Traditionally a portfolio of common stocks has been employed as a
proxy for the market portfolio. According to the CAPM, the market portfolio
should contain all risky assets such as common stocks, bonds, preferred
stocks, real estate, human capital, etc. Construction of such a portfolio
is very difficult, if not impossible, because the data on thesa assets are
not readily available.

A review of the literature on bonds reveals that different proxies
for the market portfolio have been employed. Percival [27] and McCalium
[21] used an equally weighted portfolio of their bonds. Friend and Blume
[14] and Warner [34] utilized a common stock portfolio, while Reilly and
Joehnk [28] used three different bond portfolios and two different common
stock portfolios. As demonstrated by Roll [29] the choice of the market
portfolio greatly affects the estimated beta. In this study a value
weighted market portfolio is constructed which includes common stocks,
corporate bonds and government bonds each weighted by their corresponding

-7-

market value. We believe this is a more reasonable proxy for the
market portfolio and clearly superior to the proxies employed in other
studies.

C. STATISTICAL TECHNIQUES

Since the stationarity of beta is a time related phenomenon, the
traditional method of testing for stationarity using correlation coeffi-
cients is inappropriate. There are basically two problems with the use
of the correlation coefficient as a measure of stationarity. First,
when using equation (1) to estimate 3, it is implicitly assumed that g
is stationary during the estimation period. Second, the correlation
coefficient cannot be used to determine the stationary of the individual
securities. It is, in essence, an aggregate measure of stationarity of
the betas for a group of aecurities or portfolios. An ideal test for
stationarity should detect the constancy of the security beta over time
by examining whether or not the regression coefficients in the market
model vary over time.

Since we were primarily interested in the stability of S. (not
o, and e. simultaneously) we also estimated g. by;

^it == ^iV ^ ^it* ^^>

where r^^ = R^^ - R^^, r^^ - R^^ - R^^, R^^ is the risk free rate of

7
interest and the intercept (a.) was supressed. To correctly examine

the behavior g, over time, equation (6) is rewritten as;

y, = e, x^ + e, (7)

-8-

where subscript t on 3 indicates that it may vary over time, y is
the vector of returns on a bond, x is the vector of returns on the
market portfolio, and e is the vector of disturbances. The null
h3rpo thesis for stationarity is formulated as:

Hq! Bj^ = 32 " ••• = ^T* ^^^

In words, the null hypothesis states that 3 is stable over time. The
alternate hypothesis is that not all 3's for an individual bond are
equal .

The stationarity of 3 problem is a special case of the general class
of problems concerned with detection of changes in the regression model
structures over time. Early work on detecting changes in a model struc-
ture employed the ordinary least square (OLS) residuals or the cumulative
sum of the OLS residuals. The difficulty with these approaches, however,
is that there is no known method of assessing the significance of the
nonstationarity in the regression coefficients (cf., Mehr and McFadden
[22]). To avoid problems associated with the OLS residuals. Brown and
Durbin [6], and Brown, Durbin, and Evans (BDE) [7] proposed using recur-
sive residuals. BDE have shown that under the null hypothesis of station-
arity the recursive residuals have the desirable properties of being
uncorrelated, with zero mean and constant variance, and therefore are
Independent of each other under the normality assumption. Recursive
residiials are also preferrable to OLS residuals for detecting nonstation-
arity in 3 because until a change takes place the recursive residuals
behave exactly as specified in the null hypothesis. Recursive residuals
are defined as:

-9-

w « (y -x'b -)/[! + x' (X' ,x j"-^x 3*^ (9)

r ^-^r r r-1' ■■ r r-1 r-1 r

r = kt-l, ..., T

where k Is the number of regression coefficients (1 in equation (7)),
^r-1 " [x^.---.\_iK \ = (x;x^)~^x;y^, and Y^ = (y^ y^ .

For each value of r, which in our study takes a value between 2 and 72,

g
the recursive residual was computed using equation (9) .

BDE derived a statistical test for stationarity using the cumu-
lative sum of the squared recursive residuals. This test, the cusum
of squares test, detects both systematic and random changes in the 3
and is based on the following formulation:

r T

s = ( Z w^)/( Z w^), r=k+l T. (10)

^ j=lH-l ^ j-'lcfl ^

Under the null hypothesis s has a beta distribution with mean

(r-k)/(T-k). BDE suggested constructing a confidence internal for s as

[(r-k)/(T-k)] + C where C is chosen from Table 1 of Durbin [9]. The

stationarity hypothesis will be rejected if |s -((r-k)/(T-k)) |>C for

any r Included in [fc4-l,T].

If 6 is expected to change systematically over time another test

can be used to detect such changes [7]. This type of nonstationarity can

be tested using an F-test. Under the null hypothesis of stationarity,

equation (7) can be rewritten as:

where 3^ denotes that the beta coefficient is stationary. Equation (11)
is the reduced model; under the alternate hypothesis 3 is assumed to
change linearly with time, or

-10-

y, - x; 3^ + e^. ^ (12)

where ^t " ^0 "*" ^'^* ^°^ ^^'^^

where 6 Is the coefficient of time. Substitution of equation (13) into (12)
yields:

^t ' ""t^^O "^ ^'^^ "^ ^t» ^^'^^

which is the full model. The null hypothesis of stationarity is tested
by a comparison of the mean-square increase in the explained variation
with the error variance. This F-test is:

SSE(R)-SSE(F) . SSE(F)
^ ' df(R)-df (F) ^ df (F) (15)

where SSE(R) and SSE(F) are the error sum of squares of the reduced and
full models, respectively. Likewise, df(R) and df (F) are the degrees of
freedom associated with the SSE(R) and SSE(F). It should be noted that
this F-test detects only systematic changes in g, whereas the cusum of
squares test detects both systematic and random changes in 3. In this
study the nonstationarity detected by the F-test is called "systematic
nonstationarity", while the nonstationarity detected by the cusum of squares
test but not with the F-test is called "random nonstationarity".

IV. EMPIRICAL RESULTS
A. SAMPLE CHARACTERISTICS

In Table 1 the sample characteristics are reported broken down by
industrial versus public utility bonds. In general, the coupon rates
are lower for the industrial bonds as are the years to maturity while

-11-

TABLE 1
Characteristics of the Sampled Bonds

TOTAL

INDUSTRIAL

Number

84

42

PUBLIC UTILITY 42

Coupon
Rate

6.452 .

Issue Years to
Size Maturity 3^

60.071

27.274

.410 .181 .423

(.638) (48.646) (3.671) (.187) (.126) (.178)

6,129 81.905

(.564) (65.490)

6.775 38.238

(.541) (41.194)

25.714 .412 .177 .428

(2.361) (.245) (.154) (.232)

28.833 .407 .185 .419

(4.090) (.104) (.092) (.102)

Standard deviation in parethesis.

In millions of dollars.
'^From the market model given by equation (1) .
^rom the market model given by equation (6).

-12-

the Issue sizes are larger for the Industrial bonds than for public
utility bonds. These findings are consistent with the typical charac-
teristics of public utility bonds which have higher coupon rates and

longet maturities. The average 3 of 84 bonds obtained from equation

2
(1) is 0.410 with an average R of 0.181. The average 3 of the bonds

obtained by employing equation (6) is .423 (where a, was suppressed)

which is virtually the same as that obtained by using eqxiation (1) .

There are no significant differences in bond betas between the public

9
utility and industrial groups. In the rest of the study $. as

estimated by equation (6) is employed.

B. STATIONARITY OF SYSTEMATIC RISK FOR BONDS

The cusum of squares test (for random nonstationarity) and the F-test
(for systematic nonstationarity) were applied to each bond to detenoine
whether the individual bond betas were stable or not over the period
examined. The results of these tests (employing a 5 percent significance
level) are reported in Table 2. Examination of this table indicates
that 69.05 percent [(24+5)/42] of the industrial bonds had nonstationary
betas, while 95.24 percent [ (24+16) /42] of the public utility bonds had
nonstationary bond betas. Overall, 82.14 percent [ (48+21) /84] of the
bonds examined had nonstationary betas with 25 percent (21/84) of the
bonds exhibiting systematic nonstationarity and 57.14 percent (48/84)
indicating random nonstationarity. Not only were more of the public
utility bond betas unstable, but they also exhibited more systematic
nonstationarity than did the Industrial bonds. These results, for a ^
very homogeneous set of bonds, provide strong empirical support for the

rO*-;Kv .«.;.^! J:: =

(l9V»l

•>i'.. ' 0

io;. i.^>,

-13-

TABLE 2

Niimber of Bonds With Nonstatlonary Beta
Based on the Cusum of Squares and F Tests
(5 percent significance level)

INDUSTRIAL

13

PUBLIC UTILITY

2

TOTAL

15

Nonstatlonary
Stationary Random Systematic Total

24 5 42

24 16 42

48 21 84

-14-

theoretlcal considerations presented in section II indicating that bond
betas are inherently nonstationary.

Since our concern is not with the nature of the nonstationarity,
per se, the rest of the analysis will focus on two groups of bonds — those
with stationary betas and those with nonstationary betas (encompassing
both random and systematic nonstationarity) . In Table 3 the salient char-
acteristics of these two groups of bonds are presented. As expected
(based on Table 1 and the knowledge that more public utility bonds are
Included in the nonstationary group), the nonstationary bonds had a
significantly higher average coupon rate, significantly smaller average
size and significantly lower betas than bonds with stationary betas.
While not statistically significant (at the 5 percent level), the bonds
with nonstationary betas tend to have slightly lower bond ratings, while
there is virtually no difference in the average years to maturity. The
higher average coupon rate for bonds with nonstationary betas can also
be seen by examining Table 4. Almost 50 percent (34/69) of the bonds
with nonstationary betas have coupon rates greater than 6.5 percent while
only 13 percent (2/15) of the bonds with stationary betas have coupon
rates greater than 6.5 percent.

As presented in Section II, theoretical considerations indicate
bond betas should be inherently unstable and this instability is related
to: 1) the duration of the bond, D, ; 2) the correlation between the
change in the yield to maturity of the bond and the return on the market,
-p(dy. ,R ); and 3) the standard deviation of the change in the yield ^
to maturity of the bond relative to the standard deviation of the return
on the market, a(dy. )/a(R ). (As indicated in equation (4) the

X'j

'*yi^.

-15-

TABLE 3

Statistics on Bonds with Stationary
and Nonstationary Bond Betas

Number

Stationary
15

Nonstationary
69

F Ratio

Probability

Coupon Rate

6.005
(.577)^

6.550
(.612)

9.93^

.0023

Years to Maturity'^

19.267
(4.399)

20.043
(3.771)

.91

.3439

Issue Size

110.000
(73.969)

49.217
(49.001)

16.61

.0001

Bond Rating®

2.333
(.900)

2.696
(.845)

2.21

.1407

Bond Beta

.535
(.166)

.399
(.173)

7.83

.0064

Standard deviation in parenthesis.
With 1 and 82 degrees of freedom.
'As of January 1, 1975.

In millions of dollars.

'1 = Aaa/AAA, 2 = Aa/AA, 3 = A/A and 4 = Baa/BBB.

-16-

TABLE 4

Coupon Rates for Bonds with
Stationary and Nonstationary Bond Betas

COUPON RATE

< 5.5

5.51
to 6.00

6.01
to 6.5

6.51
to 7.00

7.01
to 7.50

> 7.50

TOTAL

Stationary
Nonstationary

3

4

6

14

4
17

1
16

1
17

1

15
69

-17-

relatlonship between 3. and these factors carries a negative sign — for
convenience we have appended the negative sign to the correlation.) In
order to examine the relative Impact of these factors on the observed
stability/instability of the bond betas, we arbitrarily divided the study
period into three 24 month periods. Then we calculated the average dura-
tion, Djj.* the average correlation, p(dy. ,R ) and the average
standard deviation, a(dy. ) for the first and last 24 month periods
and the relative change in these variables from the first to the last
period. (Since the standard deviation of the market, a(R ) is the same
for all bonds, we ignore it and focus solely on o(dy. ),

The results of this analysis of the change in duration, correlation
and standard deviation for the two groups of bonds are presented in Table
5. For the bonds with stationary betas, the duration decreased, the
standard deviation in the yield to maturity increased, and the correla-
tion between the change in the yield to maturity and the return on the
market portfolio decreased from the first to the third 24 month period.
The same directional changes occurred for the bonds with nonstationary
betas. However, the Important difference in the two groups of bonds is
the relative change (columns (3) and (6) of Table 5) in these three
variables for the two bond groups.

Starting with duration. Table 5 indicates that the relative change
in duration between bonds with stationary or nonstationary betas are
approximately the same. Hence, differences in average duration are not
significant In dlfferrentiating between bonds with stationary versus
nonstationary betas (given the relatively homogeneous maturity of the
bonds under study) .

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-19-

Moving to the changes in the standard deviation of the change in
the yield to maturity, the instability of the change in the yield to
maturity increased for both groups of bonds. (This is to be expected
because of the shorter average maturity of all bonds in the last period
relative to the first period. In addition, the wider dispersion In
corporate bond returns in the last period relative to the first
period may also contribute to the increase in the observed standard
deviations.) However, the important point concerning the standard de-
viations is that the standard deviation of the nonstationary bonds
increased relatively more (1.291 to 1.064) than for bonds with stationary
bond betas. We believe the primary reason for the higher relative stan-
dard deviation for the bonds with nonstationary betas is due to the higher
coupon rates and associated higher yields to maturity for the nonstationary
bonds. (Not only do the nonstationary bonds have higher average coupon
rates, but they also have lower average bond ratings. It is well known,
ceteris paribus that the yield to maturity on lower rated bonds are larger
than for higher rated bonds.) As interest rates in general fluctuate, the
changes in the yield to maturity is larger for the nonstationary bonds
(which have higher average coupon rates and lower bond ratings); hence,

they have larger relative standard deviations than bonds with stationary

12
betas. Thus, the most important factor identified in this study

which differentiates between bonds with stationary betas versus those

with nonstationary betas is the relative standard deviations in the changes

in the yield to maturity. Higher coupon rates and yields to maturity

(leading to larger standard deviation in the changes in the yield to

maturity) are associated with bonds having nonstationary betas.

-20-

Finally, it is noted that the correlation between the changes in
the yield to maturity and the return on the market decreased for both
stationary and nonstationary bonds from the first to the last periods.
This is as expected since the sampled bonds in the third period have
shorter maturities and hence their yields tend to move less with the re-
turns on the market which are influenced by common stock as well as bond

13
returns. While not significantly different (at the .15 level), the ab-
solute value of p(dy.^,R ^) tended to be lower over time for the non-
it mt

stationary bonds (.6875 to .8847) than for bonds with stationary betas.
Again, this difference appears to be due to the higher coupon rates and
yield to maturity carried by the nonstationary group of bonds relative
to the stationary bonds.

In order to test the overall ability of the three hypothesized fac-
tors to differentiate between bonds with stationary betas and those with

2
nonstationary betas, Hotellings T was employed. It resulted in an

F ratio of 2.22 which, with 3 and 80 degrees of freedom, has a probability

value of .091. Thus, at the 10 percent significance level the three

hypothesized factors (in combination) differentiated between bonds with

stationary betas and those with non-stationary betas.

V. SUMMARY AND CONCLUSIONS
Recently a number of researchers have attempted to employ the market
model to estimate systematic risk (i.e., beta) for bonds. In this study
we reviewed theoretical evidence which suggests bond betas can be expected
to be nonstationary. This nonstationarity is a function of the duration
of a bond, the standard deviation of the change in the yield to maturity

-21-

of a bond relative to the standard deviation of the return on the market
portfolio, and the correlation between the change in the yield to maturity
of a bond and the return on the market portfolio. However, all bonds
will not necessarily have nonstationary betas in a given time period since
it is possible that these factors may occasionally counteract one another.
Empirical tests indicated that over 80 percent of the bonds examined
had nonstationary betas. The primary factor differentiating bonds with
nonstationary betas from those with stationary betas was the substantially
higher relative standard deviation in the change in the yield to maturity
for bonds with nonstationary betas. The larger standard deviation was
caused by the higher average coupon rates and yields to maturity for bonds
with nonstationary betas. The substantial presence of nonstationarity
in public utility bond betas is caused by the peculiar nature of long term
financing in the public utility industry which results in generally higher
coupon rates and yields to maturity than in the industrial sector. The
theoretical and empirical results of this study indicate bond betas, in
general, tend to be nonstationary. Hence, further use of them appears
to be of very questionable value.

-22-

FOOTNOTES

Livingston [18] extended Boquist et al.'s work by taking into
account the duration of both the security and the market portfolio.
He shows that:

D^ P(dy,,,dR^pa(dy^^)

where D is the duration of the market portfolio and dR is the
change xn the return on the market portfolio. Since the duration of
the market portfolio (which is dominated by common stocks with infinite
maturity) does not change much over time we have chosen to work with
equation (4). The notation follows that of Boquist et al. [4] and
Livingston [18] except y, , rather than r. , is used for the yield
to maturity,

2
To provide some empirical evidence for the proposition that

-p(dy. ,R ) is smaller for shorter-term bonds than for longer-term

bonds we computed -p(dy. ,R ) using basic yields on corporate bonds

with 1, 5, 10, and 15 years^'to maturity. Over the time period of 1941-

1970 the value of -p(dy. ,R ) are .47, .52, .55 and .56 for bonds

with 1, 5, 10, and 15 years'^'to maturity, respectively. Therefore, as

expected, -p(dy, ,R ) becomes smaller the shorter the term to

maturity.

3
The requirement that the bonds be consistently rated (without

any change in rating) insures that the relative risk of default (as per-
ceived by the two major rating agencies did not change over the time
period employed. Thus, even though the bonds are not default free as
required by the Boquist et al.'s model presented in equation (4), the
relative probability of default was held constant.

Recent theoretical work by Merton [23], Black and Cox [3] and
Brennan and Schwartz [5] suggests that subordination or specific bond
Indenture provisions influence the value of bonds. Subordination is not
a problem since all bonds selected for this study are non-subordinated.
In addition, an examination of the call provision indicated that the
vast majority of Issues required a five year delay if they were to be
called for refunding at a rate appreciably lower than the bond's coupon
rate. Given the general rise in interest rates during this time period
there was no economic incentive to refund. Finally, virtually all of
the industrial bonds and a small portion of the public utility bonds are
debentures. While some minor differences in the characteristics of the
bonds examined exist, there is no reason to believe that any systematic
tendencies are present which influence the results.

-23-

A list of 84 bonds Is available from the authors. The primary
source of the monthly price data (for the period May 31, 1969 through
May 31, 1975) was the Bank and Quotation Record [1]. Secondary sources in-
cluded Commercial and Financial Chronicle [8], Moody's Bond Record [24]
and Standard and Poor's Bond Guide [321. The closing bid or sale price
was employed; however, it occasionally became necessary to use an opening
ask price. The availability of data was less of a problem for the public
utility bonds than for the industrials in that closing bid or sale prices
were almost uniformly available for the public utility issues examined.
Other features of the bonds were deteirmined by reference to Moody's Public
Utility [26] and Moody's Industrial [25] manuals.

The common stock returns employed were those from the CRSP value-
weighted index while the corporate and government bond returns were those
(as updated) provided by Ibbotson and Sinquefleld [16]. The common stock
weights employed were obtained from the Statistical Bulletin [33] while
the corporate and government bond weights were obtained from the Economic
Report of the President [11]. It can be shown that the use of a common
stock index for R will result in lower estimated bond betas. We
conducted part of™the analysis with the CRSP values-weighted index— there
were no significant differences between those results and the reported
findings.

We also examined the statlonary/nonstationarlty of a. and
3. simultaneously as estimated by equation (1) . The subsequent find-
ings are virtually the same whether we focus on the statlonarity of both
a. and 3^ as estimated by equation (1) or only the statlonarity
or 3j as estimated by equation (6) .

o

The computer program to test the statlonarity of 3 is pro-
vided by BDE [7].

9
The bond betas did vary by bond rating group with a mean of 0.570

for the Aaa/AAA group, 0.450 for the Aa/AA group, 0.391 for the A/A group

and 0.372 for the Baa/BBB group. A one-way analysis of variance yielded

an F-ratlo of 2.87 which, with 3 and 80 degrees of freedom, was significant

at the .041 level. Schwendlman and Pinches [30] reported that mean common

stock betas Increased as bond ratings decreased; our results indicated

that bond betas decrease as bond ratings decrease. While the instability

of the bond betas casts serious doubt on the interpretablllty of bond

betas, there appears to be no consistency between bond betas, common stock

betas and bond ratings. No other material differences are noted in the

sample.

Since duration changes each period, we calculated duration at
the middle of the first time period (month 12) and the middle of the
third time period (month 60).

'xhe standard deviation of returns on corporate bonds, using
the Ibbotson and Sinquefleld [16] data, was .0290 for the first time
period and .0309 for the last time period. Hence, bond returns in gen-
eral were more volatile in the last time period.

-24-

12

As an example of the relationship between bond ratings and stan-
dard deviation of the change in yield to maturity, weekly yields to
maturity were gathered for Standard and Poor's AAA., AA, A and BBB in-
dustrial and public utility bonds from July through December 1977. The
standard deviations of the change in yield to maturity for the four
bond groups over that time period were:

Industrial— AAA - .0393, AA - .0428, A - .0510, BBB - .1962; and
Public Utility— AAA - .0415, Aa - .0422, A - .0441, BBB - .0527. In all
cases the standard deviation in the changes in the yield to maturity in-
crease as the bond ratings decrease.

13

See footnote 2.

-25-

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