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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;!"'!

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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

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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

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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;.:

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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

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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:

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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

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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

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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).

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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

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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.

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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

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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.

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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.

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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.

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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.

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BIBLIOGRAPHY

1. Bank and Quotation Record. New York: William B. Daner Co., 1969-1975.

2. F. Black. "Capital Market Equilibrium with Restricted Borrowing," Journal of Business, Volume 45 (July 1972), pp. 444-454,

3. F. Black and J. C. Cox. "Valuing Corporate Securities: Some Effects of Bond Indenture Provisions," Journal of Finance, Volume 31

(May 1976), pp. 351-367.

4. J. A. Boquist, G. A. Racette and G. G. Schlarbaum. "Duration and Risk Assessment for Bonds and Common Stocks," Journal of Finance,

Vol. 30 (December 1975), pp. 1360-1365.

5. M. J. Brennan and E. S. Schwartz. "Savings Bonds, Retractable Bonds and Callable Bonds," Journal of Financial Economics, Vol. 5 (August, 1977), pp. 67-88.

6. R. L. Brown and J. Durbin. "Models of Investigating Whether a Regression Relationship is Constant Over Time," Selected Statistical Papers, European Meeting, Mathematical Centre Tracts Number 26, Amsterdam,

1968.

7. R. L. Brown, J. Durbin and J. M. Evans. "Techniques for Testing the Constancy of Regression Relationships Over Time," Journal of the Royal Statistical Society, Vol. 27 (1975), pp. 149-192.

8. Commercial and Financial Chronicle, Various Issues.

9. J. Durbin. "Tests for Serial Correlation in Regression Analysis Based on the Periodogram of Least Squares Residuals," Blometrika, Vol. 38 (1968), pp. 1-15.

10. R. M. Duval 1 and J. M. Cheney. "The Relationship of Bond Yi,elds to Capital Structure When Bankruptcy is Possible." A paper presented at the Financial Management Association Meeting, Seattle, 1977.

11. Economic Report of the President, United States Government, Washington, D.C., 1978, p. 337.

12. A. R. Eddy. "Interest Rate Risk and Systematic Risk: An Interpretation," Journal of Finance, Vol. 33 (May 1978), pp. 626-630.

13. L. Fisher. "An Algorithm for Finding Exact Rates of Return," Journal of Business (January 1966), pp. 111-118.

14. I. Friend and M. E. Blume. "The Demand for Risky Assets," American Economic Review, Vol. 65 (December 1975), pp. 900-922.

.-.Olc,

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15. M. H. Hopewell and G. G. Kaufman. "Bond Price Volatility and Term to Maturity: A Generalized Representation," American Economic Review, Vol. 63 (September 1973), pp. 749-753.

16. R. G. Ibbotson and R. H. Sinquefield, "Stocks, Bonds, Bills and Infla- tion: Year-by-Year Historical Returns (1926-74)," Journal of Business, Vol. 49 (January 1976), pp. 11-47.

17. J. Lintner. "The Valuation of Risk. Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budget," Review of Economics and Statistics, Vol. 47 (Feburary 1966), pp. 13-37.

18. M. Livingston. "Duration and Risk Assessment for Bonds and Common Stocks: A Note," Journal of Finance, Vol. 33 (March 1978), pp. 293-295.

19. F. R. Macaulay. "Some Theoretical Problems Suggested by the Movements of Interest Rates, Bond Yields and Stock Prices in the United

States Since 1856," New York: National Bureau of Economic Research, 1938, pp. 45-53.

20. B. G, Malkiel. The Term Structure of Interest Rates. Princeton, 1966.

21. J. S. McCallum. "The Expected Holding Period Return, Uncertainty and the Term Structure of Interest Rates," Journal of Finance, Vol. 30 (May 1975), pp. 307-323.

22. C. B. Mehr and J. A. McFadden. "Certain Properties of Gaussian Processes and Their First Passage Time," Journal of the Royal Statistical Society, Series B, Vol. 27 (1965), pp. 505-522.

23. R. C. Merton. "On the Pricing of Corpoorate Debt: The Risk Structure of Interest Rates," Journal of Finance, Vol. 29 (May 1974), pp. 449-470.

24. Moody's Bond Record. New York: Moody's Investors Services, 1969-1975.

25. Moody's Industrial Manual. New York: Moody's Investors Services, 1968-1976.

26. Moody's Public Utility Manual. New York: Moody's Investors Services, 1968-1976.

27. J. Percival. "Corporate Bonds in a Market Model Context," Journal of Business Research (October 1974), pp. 461-468.

28. F. K. Reilly and M. Joehnk. "The Association Between Market- Determined Risk Measures for Bonds and Bond Ratings," Journal of Finance, Vol. 31 (December 1976), pp. 1387-1403.

29. R. Roll. "A Critique of the Asset Pricing Theory's Test Part I: On Past and Potential Testability of the Theory," Journal of Financial Economics, Vol. 4, No. 2 (March 1977), pp. 127-176.

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30. C. J. Schwendiman and G. E. Pinches. "An Analysis of Alternative Measures of Investment Risk," Journal of Finance, Vol. 30 (March 1975), pp. 183-200.

31. W. F. Sharpe. "Capital Asset Prices: A Theory of Market Equilibrium Under Conditions of Risk," Journal of Finance, Vol. 19 (September 1964), pp. 425-442.

32. Standard & Poor's Bond Guide. New York: Standard & Poor's Corp., 1968-1975.

33. U.S. Securities and Exchange Commission. Statistical Bulletin, Vol. 36 (June 1977), p. 9. Washington: U.S. Government Printing Office.

34. J. B. Warner. "Bankruptcy, Absolute Priority and the Pricing of Risky Debt Claims," Journal of Financial Economics, Vol. 4 (May 1977),

pp. 239-276.

35. J. B. Yawitz. "The Relative Importance of Duration and Yield Volatility on Bond Price Volatility," Journal of Money, Credit and Banking,

Vol. 9 (February 1977), pp. 97-102.

36. J. B. Yawitz and W. J. Marshall. "Risk and Return in the Government Bond Market," Journal of Portfolio Management (Summer 1977),

pp. 48-52.

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