- Email: [email protected]
The interest rate sensitivity of equity prices with respect to systematic risk and leverage
J BUSN RES 1987:15:85-92
85
The Interest Rate Sensitivity of Equity Prices with Respect to Systematic Risk and Leverage Larry J. Johnson University of Tulsa
John R. Brick
Kelly Price
Michigan State University
Wayne Stale University
This study extends the previous empirical research into the sensitivity of equity prices to include interest rates. Specifically, the null hypothesis that the interest rate sensitivity of equity prices is independent of the level of systematic risk and financial leverage is tested. The hypothesis is tested using a short-term and longterm interest rate index. The results show that the interest rate sensitivity of equity prices is independent of the amount of financial leverage but not independent of the level of systematic risk.
Introduction Historically, empirical research into the structure of equity prices has tended to relate the firm-specific variable of earnings or expected earnings with share prices [l, 4, 9, and 111. However, more recently, financial managers and academics have seen the wisdom of including an interest rate sensitivity factor in the market models that characterize the demand for speculative assets. Stone [ll] notes that a market model consistent with the practices of money managers needs to view the investment process as being subject to the gains and losses resulting from interest rate changes, as well as market effects. Since money managers often determine the amounts to be invested in the bond and equity portions of their portfolios before choosing the particular bonds and equities that make up the portfolio, Stone suggests a twofactor market model to describe the returns on speculative assets. Two recent studies by Lloyd and Shick [7] and Lynge and Zumwalt [8] examined the interest rate sensitivity of equities by focusing on the 30 industrials that compose the Dow Jones Industrial Average. Lloyd and Shick provide evidence that an equity
Send correspondence to Larry J. Johnson, 600 South
College
Avenue,
Tulsa,
Journal of Business Research 15. 85-Y2 (1987) 0 Elsevier Science Publishing Co., Inc. 1987 52 Vanderbilt Ave., New York, NY 10017
Oklahoma
College 74104.
of Business
Administration,
University
of Tulsa.
014X-2963/87/$3.50
86
.IBUSN
L.J. Johnson et al.
RES 1987:15:85-92
index is a significant variable in explaining the returns of the DJIA and that the bond index increases the explanatory power of the market model in most cases. However, Lloyd and Shick used Salomon Brothers’ Total Performance Index for the High-Grade, Long-Term Corporate Bond Market rather than a range of maturities as suggested by Stone [ll]. Lynge and Zumwalt, in the spirit of Lloyd and Shick, then examined a multi-index market model that introduced both a longterm and short-term interest rate index in conjunction with an equity index. They also used the DJIA as a sample for testing purposes. They concluded that the single-index market model may be improved by including a debt return index for a large number of firms in nonfinancial industries. Both of these studies demonstrate that a significant extra-market force exists in relation to the Sharpe-Lintner security market line. This paper is concerned with the implications this has for the Sharpe-Lintner security pricing theory. This paper is designed to test the hypothesis that the interest rate sensitivity of a common stock portfolio is independent of the portfolio’s systematic risk and financial leverage.
The
Market
Models
The Sharpe-Lintner security market line describes a security pricing theory based on systematic risk, whereby systematic risk is the comovement of security returns with a market return index. Unsystematic risk does not affect the return of a security, assuming investors hold diversified portfolios. This has also been termed the security’s specific risk. This asset pricing model asserts that systematic risk is the sole determinant of asset prices. Symbolically, the return-generating model can be represented by the now familiar market model, i.e., R,., = a, + b,%,,
+ Q,
(1)
where, l?,,.,is the return
on equity
I?,,, is the return
on an equity
e,,, is the unsystematic
equity
a, and b, are security
i, in time t,
market return
index, on security
j,
specific constants,
b, is the Cov (R,, R,) R, is the return
security
/ Var (R,),
on a risk free asset,
a, is the R, (1 -b,), ER,., = a, + b, ER,,,, 2 a/ = b; cr: + a:, b, = b’, (1 + (1 - t) (DIE),) where, b’, is the systematic t is the corporate and
risk of an unlevered tax rate
equity
security,
Interest-Rate
J BUSN RES 1987:15:85-92
Sensitivity (DIE),
is the leverage
ratio of corporation
j.
In contrast, Stone’s [ 1 l] security market line is based on a security-pricing theory that not only prices systematic equity-risk but also systematic interest rate risk. Now, however, a diversified portfolio must include a bond portion that represents more than one maturity range in order to diversify and minimize the effect of a change or twist in the term structure of interest rates. Symbolically, the returngenerating model can be represented by a multi-index market model like the one purposed by Lynge and Zumwalt [8], i.e., R,,, = A, + B, d,,,,, + C,%., + Dj%., + &,,
(2)
where &
is the return
R,,, is the return
on a short-term
bond,
on a long-term
l?,,., is the unsystematic
equity
return
A,, B,, C,, and D, are security ER,,, = Aj + B, ER,,,
bond, on a security
j,
specific constants,
+ C, ER,,,
+ 0, ER ,.,,
and a; = B; a: + CT a: + D; o: + uzE. A disciple of a the Sharpe-Lintner (2) as a special case of (l), where
security market in the following
line would view market four conditions hold,
model
I) A, = a,, 2) Aj + BIER,.,
+ C,ER,,,
+ D,ER,,,
= aj + biER,,,,
3) B; u,f + C: at + Df a: = by ai, and 4) 0: = a:. Based on the findings of the previous research concerning the interest rate sensitivity of equity prices [6, 7, 8, and 111, one can conclude that stock prices are interest rate sensitive. It is the primary concern of this study to determine if the implicit treatment of a portfolio’s interest rate sensitivity in the return-generating model, eq. (1) is a source of systematic bias in portfolio return characterizations. If a portfolio’s interest rate sensitivity is independent on the portfolio’s systematic risk and leverage, then the implicit treatment of the interest rate sensitivity in a portfolio-return-generating model such as model (1) does not bias portfolio return characterizations. Using a time series test for the independence of two random variates, (see Appendix A) the independence, or lack thereof, between both a short-term and long-term interest-rate index and each of ten portfolio price indexes will be examined to determine if the interest rate sensitivity of a portfolio price index is systematically related to the portfolio’s systematic risk or leverage. The data set used in this study will be described in the next section. Construction
of Stock
Price
Indices
and
Interest
Rate
Series
The stock price indices were formed from firms that are included on both the CRSP and COMPUSTAT data bases; they have December fiscal year ends and complete
88
.I BUSN
RES 1987:15:85-92
L.J. Johnson et al.
data over the interval l/74 to 12/78. This screening process netted 726 firms. These firms were then divided into five portfolios based on median debt ratios over the period. Each of these five portfolios were then divided into two portfolios based on the betas of their included firms. Each of the resulting ten portfolios then contained 72 or 73 securities. A stock price index was constructed for each of the ten portfolios, assuming that the firms included in the index remained constant throughout the period. Each index was equally weighted, with rebalancing occurring at the end of each month. The data source for the two interest rate time series is the H-15 release of the Federal Reserve. This release is published weekly, and contains interest rate observations for each trading day of the prior week. The observations were taken from the last trading day of each month. The next section reports the time series statistics i,,(z,n), ?,lk(~,n), SZ(n,q), and Sz(n,k) as described in Appendix A.
Conclusions
and
Implications
The major conclusion of this study deals with the hypothesis developed earlier. This paper demonstrates that Sharpe’s systematic risk measure may be a biased measure of portfolio returns with respect to interest rate sensitivity. This inference can be drawn by examining the test results. Note that the portfolio’s interest rate sensitivity, or lack thereof, follows a pattern with respect to p (6,) but not with respect to the leverage employed by the firms in the portfolio. The interest rate sensitivity of portfolio prices with respect to the short-term interest rate was found to be similar for both high and low systematic risk portfolios. However, the highrisk portfolio price’s have been shown to be independent of the long-term interest rate while the low-risk portfolio price’s are sensitive to the long-term interest rate. In general, the interest rate sensitivity of low-risk portfolios is greater than that of high-risk portfolios because the latter are sensitive to both the short-term and long-term interest rates, whereas the high-risk portfolios are sensitive to only the short-term interest rate. This implies that a return-generating market model should include more than just an equity market index. In addition it should include a debt index that represents a range of maturities or a set of debt indices that represent different maturities, i.e.,
R,,,= A, + B,&, +
C$,,,
+
D,&
+ E,,,.
Noting the caveat concerning the interpretation of the i,,, (z,n) and i,, (z,n) presented in Appendix A, if we assume that the cross correlations are normally distributed with standard errors equal to [1.96(T - I z I) “‘I, there is evidence that interest rates lead stock prices (see Table 1 and Table 2). It is clear from the significance of coefficients for a lead of one, as well as the sign distributions over the lag/lead range and trends in coefficients, that there is evidence that interest rates tend to lead stock prices. The most significant coefficients are at a lead of one period for three-month bill rate changes for both low- and high-beta portfolios. For long-term bond rate changes there is weaker evidence of significance at a leadof-one period for both low- and high-beta portfolios. This suggests that a next step in analyzing the interest rate sensitivity of equity prices should be along the lines of a “causality” or “lead/lag” time series test.
Interest-Rate
J1987:15:X5-92 BUSN RES
Sensitivity
Table 1. Cross Correlations: January
1974 to December
Three-Month 1978
Bill Rate Changes
89
and Stock Price Changes-
Low Beta Portfolios
Lag
Low Debt
Med-lo Debt
Medium Debt
Med-hi Debt
High Debt
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
0.05 0.16 0.23 -0.08 0.29 0.04 -0.04 -0.40 -0.02 -0.08 -0.14 -0.18 0.12
0.11 0.14 0.17 -0.03 0.17 0.03 -0.09 -0.41” 0.09 -0.05 -0.17 -0.17 0.23
0.16 0.15 0.08 0.03 0.13 -0.00 -0.13 -0.48 0.07 -0.04 -0.25 -0.14 0.23
0.10 0.11 0.16 -0.07 0.05 0.06 -0.22 -0.57” 0.12 -0.07 -0.23 -0.09 0.16
0.24 0.10 -0.01 0.13 0.13 0.04 -0.23 -0.35” 0.03 -0.03 -0.24 -0.13 0.18
24.95
24.39
28.19
32.76”
23.76
S,=
d.f. = 13
High Beta Portfolios
Lag
Low Debt
Med-lo Debt
Medium Debt
Med-hi Debt
High Debt
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
0.25 0.08 0.14 0.03 0.25 0.00 -0.22 -0.40 0.10 -0.18 -0.14 -0.05 0.10
0.08 0.20 0.18 0.14 0.10 0.12 -0.29 -0.32 0.02 -0.08 -0.26 -0.02 0.16
0.17 0.24 0.09 0.12 0.16 0.11 -0.28 -0.31” 0.02 0.03 -0.27 -0.05 0.16
0.18 0.04 0.20 0.08 0.18 0.03 -0.18 -0.41” -0.00 -0.07 -0.25 -0.03 0.11
0.12 0.16 0.28 0.05 0.22 0.06 -0.18 -0.36” -0.06 0.06 -0.27 0.02 0.13
s;, =
26.98
24.94
26.92
24.12
26.20
d.f. = 13
“Indicatessignificanceat 0.05 level.
References 1. Beaver, William. The Informational Content of Annual Earnings Empirical Research in Accounting: Selected Studies. 1968 Supplement Journal of Accounting Research (1968). Box, G. E. P. and Jenkins, G. J. Time Series Analysis: Francisco: Holden-Day, Inc., (1970).
Forecasting
Announcements. to Vol. 6 of the and Control.
San
Box, G. E. P. and Pierce, D. A. Distribution of Residual Auto-Correlations in Autoregressive-Integrated-Moving Average Time Series Models. Journal of the American Statistical Association. 65: 1509-26 (December 1970). Brown, Philip and Ball, Ray. An Empirical Evaluation Journal of Accounting Research. 6: 159-78 (Autumn Haugh, Larry D. Checking
the Independence
of Accounting 1968).
of Two Covariance
Income
Stationary
Numbers.
Time Series:
90
BUSN RES 1987:15:85-92
J
L.J. Johnson
Table 2. Cross Correlations: Long-Term Government Changes-January 1974 to December 1978
Rate Changes
et al.
and Stock Price
A. Low Beta Portfolios Low Debt
Med.10 Debt
Medium Debt
Med-hi Debt
High Debt
0.01 0.28 -0.01 0.05 0.06 0.13 -0.23 -0.30” 0.13 -0.07 -0.13 -0.19 -0.04 19.84
0.03 0.19 -0.08 0.11 0.02 0.16 -0.38” -0.16 0.29” -0.11 -0.15 -0.08 -0.02 22.40
0.05 0.14 -0.15 0.05 0.04 0.14 -0.41” -0.19 0.26 -0.10 -0.19 0.10 0.03 24.08”
0.18 0.08 -0.14 0.04 0.05 0.12 -0.44” -0.37” 0.22 -0.11 -0.13 -0.01 -0.07 28.76”
0.18 0.0s -0.19 0.05 0.15 0.03 -0.40 -0.28” 0.14 -0.06 -0.11 -0.14 0.08 23.97”
Lag -6 -5 -4 -3 -2 -1 0 1 2 3 4
5 6 s,=
d.f. = 13
B. High Beta Portfolios
Lag
Low Debt
Med-lo Debt
Medium Debt
Med-hi Debt
High Debt
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 s,=
0.12 0.10 -0.18 0.17 0.09 0.12 -0.31” -0.15 0.12 -0.11 -0.08 -0.11 -0.04 16.33
0.09 0.06 -0.09 0.17 0.11 0.18 -0.34 -0.18 0.16 -0.15 -0.20 -0.06 -0.05 19.95
0.10 0.07 -0.10 0.04 0.16 0.21 -0.38 -0.20 0.25 -0.13 -0.14 -0.09 -0.05 23.44”
0.10 0.05 -0.09 0.06 0.07 0.15 -0.30” -0.28” 0.19 -0.15 -0.13 -0.10 -0.05 18.41
0.17 0.07 -0.05 0.08 0.12 0.12 -0.28” -0.27” 0.15 -0.07 -0.18 -0.08 0.05 17.56
d.f. = 13
“Indicates significanceat 0.05 level
A Univariate Association.
Residual 71: 378-385
Cross-Correlation (June 1976).
Approach.
Journal
of the American
Statistical
6. Joehnk, Michael D. and Nielson, James F. The Effect of Interest Rates on Utility Share Prices. Review of Business & Economic Research. 12: 35-43 (Winter, 1976-77). 7. Lloyd, William P. and Shick, Richard Journal
of Financial
and Quantitative
A. A Test of Stone’s Two-Index Model of Returns. Analysis. 12: 363-76 (September 1977).
8. Lynge, Morgan J., Jr. and Zumwalt, J. Kenton An Empirical Study of the Interest Rate Sensitivity of Commercial Bank Returns: A Multi-Index Approach. Journal of Financial and Quantitative Analysis. 15: 731-42 (September 1980). 9. Malkiel, American
10. Ofer,
Burton
J. and Cragg, John G. Expectations Review. 60: 601-17 (September
Economic
Aharon
R. Investors’
Expectations
and the Structure 1970).
of Earnings
Growth:
Their
of Share Prices. Accuracy
and
Interest-Rate
J BUSN RES 1987:15:85-92
Sensitivity
Effects on the Structure of Realized Rates of Return. Journal of Finance. 30: 508-23 (May 1975). 11. Stone, Bernell K. Systematic Interest Rate Risk in a Two-Index Model of Returns. Journal of Financial and Quantitative Analysis. 9: 709-25 (November 1974).
Appendix
A
A Test for the Independence
of Two Time Series
The method used in this study to investigate the stochastic relationship between two covariance-stationary time series involves two stages. The first stage calls for fitting univariate models to each of the series Stage two entails cross-correlating the resulting two strictly stationary residual series. Haugh [5] has shown that under the null hypothesis of series independence, when the set of residual cross correlations are squared and then weighted and summed, the asymptotic distribution of this weighted average is x-square. The univariate dynamic model used in this study to describe the time series, p,,, (price index of the stock portfolios), and r,,, and r,,! (the short-term and long-term interest rate) is a nonlinear, rational-distributed lag model. Statistically, this model has been described by Box and Jenkins [2]. To fit such a univariate model Box and Jenkins suggest the three-step process of 1) specification of the models form, 2) estimation of the parameters of this model by a nonlinear, least-squares, maximum-likelihood method and 3) adequacy testing to determine the models fit. Assume that @,,n: t = 1,2, . . . $0 and n = 1,2, . . ,lO}, {T,,~:r = 1,2, . . . ,60} and by a mixed auto-regressive{r,.j t = 1,2, . . . ,60} can be modeled individually integrated-moving average model, i.e.,
fn(B)(l - qp,,, = g,(wL,,, n = 12,. . . 710,
(4)
and i(B)(l v(B)(l
- B)r,,, - Q,.,
= j(B)&, = m(B)&
(5)
normal series. Using the specification and where - denotes a strictly stationary estimation technique provided by Box and Jenkins, the parameters (f,(B), g,(B), (these results will be provided upon request). i(B), v(B), m(B)] were estimated The Box-Pierce [3] test statistic was used to examine the adequacy of the fitted models (Only two of the fitted models were not strictly random-walk or firstdifference models. The low-risk and low debt-to-equity price index and r,,, demonstrated the need for a low-order moving average component that was removed, resulting in a strictly stationary, residual-time series). The next stage in testing for the independence of two time series is to calculate the cross correlations between the residuals from the univariate models. Specifically, the cross correlations of the ten portfolio residuals with respect to both the short-term and long-term interest-rate residuals were calculated as follows, i.e.,
92
J BUSN RES 1987:15:RS-92
L.J. Johnson
et al.
where n = 1,2, . . . ,lO; z = 6,5, . . . , - 5, -6; and - denotes that these variables are the residual estimates of the true white-noise sequences (These residual cross correlations, fh,,(z,n) and ?,&,n) are reported in Tables 1 and 2). It is important to note that the individual i,,(z,n) and ?,,,,(z,n) can’t be examined for significance since the true parameter values are not known and hence the covariance matrix of the individual cross correlations are unknown. However, Haugh [5] has shown that the following test statistic, S$ is approximately distributed as x-square with 2M + 1 degrees of freedom under the assumption of series independence, i.e., S;(q,n)
= T *
c
(T - I z
I)-‘f&(z,rz),
*=-m
(7)
$,,(/+I) = T * 2 (T - I z I)-‘%(z>& where T = 60 and m = 6. These results can be used to test the independence of two covariance-stationary series (The Sz(q,n) and Sz(k,n) are reported in Tables 1 and 2).
85
The Interest Rate Sensitivity of Equity Prices with Respect to Systematic Risk and Leverage Larry J. Johnson University of Tulsa
John R. Brick
Kelly Price
Michigan State University
Wayne Stale University
This study extends the previous empirical research into the sensitivity of equity prices to include interest rates. Specifically, the null hypothesis that the interest rate sensitivity of equity prices is independent of the level of systematic risk and financial leverage is tested. The hypothesis is tested using a short-term and longterm interest rate index. The results show that the interest rate sensitivity of equity prices is independent of the amount of financial leverage but not independent of the level of systematic risk.
Introduction Historically, empirical research into the structure of equity prices has tended to relate the firm-specific variable of earnings or expected earnings with share prices [l, 4, 9, and 111. However, more recently, financial managers and academics have seen the wisdom of including an interest rate sensitivity factor in the market models that characterize the demand for speculative assets. Stone [ll] notes that a market model consistent with the practices of money managers needs to view the investment process as being subject to the gains and losses resulting from interest rate changes, as well as market effects. Since money managers often determine the amounts to be invested in the bond and equity portions of their portfolios before choosing the particular bonds and equities that make up the portfolio, Stone suggests a twofactor market model to describe the returns on speculative assets. Two recent studies by Lloyd and Shick [7] and Lynge and Zumwalt [8] examined the interest rate sensitivity of equities by focusing on the 30 industrials that compose the Dow Jones Industrial Average. Lloyd and Shick provide evidence that an equity
Send correspondence to Larry J. Johnson, 600 South
College
Avenue,
Tulsa,
Journal of Business Research 15. 85-Y2 (1987) 0 Elsevier Science Publishing Co., Inc. 1987 52 Vanderbilt Ave., New York, NY 10017
Oklahoma
College 74104.
of Business
Administration,
University
of Tulsa.
014X-2963/87/$3.50
86
.IBUSN
L.J. Johnson et al.
RES 1987:15:85-92
index is a significant variable in explaining the returns of the DJIA and that the bond index increases the explanatory power of the market model in most cases. However, Lloyd and Shick used Salomon Brothers’ Total Performance Index for the High-Grade, Long-Term Corporate Bond Market rather than a range of maturities as suggested by Stone [ll]. Lynge and Zumwalt, in the spirit of Lloyd and Shick, then examined a multi-index market model that introduced both a longterm and short-term interest rate index in conjunction with an equity index. They also used the DJIA as a sample for testing purposes. They concluded that the single-index market model may be improved by including a debt return index for a large number of firms in nonfinancial industries. Both of these studies demonstrate that a significant extra-market force exists in relation to the Sharpe-Lintner security market line. This paper is concerned with the implications this has for the Sharpe-Lintner security pricing theory. This paper is designed to test the hypothesis that the interest rate sensitivity of a common stock portfolio is independent of the portfolio’s systematic risk and financial leverage.
The
Market
Models
The Sharpe-Lintner security market line describes a security pricing theory based on systematic risk, whereby systematic risk is the comovement of security returns with a market return index. Unsystematic risk does not affect the return of a security, assuming investors hold diversified portfolios. This has also been termed the security’s specific risk. This asset pricing model asserts that systematic risk is the sole determinant of asset prices. Symbolically, the return-generating model can be represented by the now familiar market model, i.e., R,., = a, + b,%,,
+ Q,
(1)
where, l?,,.,is the return
on equity
I?,,, is the return
on an equity
e,,, is the unsystematic
equity
a, and b, are security
i, in time t,
market return
index, on security
j,
specific constants,
b, is the Cov (R,, R,) R, is the return
security
/ Var (R,),
on a risk free asset,
a, is the R, (1 -b,), ER,., = a, + b, ER,,,, 2 a/ = b; cr: + a:, b, = b’, (1 + (1 - t) (DIE),) where, b’, is the systematic t is the corporate and
risk of an unlevered tax rate
equity
security,
Interest-Rate
J BUSN RES 1987:15:85-92
Sensitivity (DIE),
is the leverage
ratio of corporation
j.
In contrast, Stone’s [ 1 l] security market line is based on a security-pricing theory that not only prices systematic equity-risk but also systematic interest rate risk. Now, however, a diversified portfolio must include a bond portion that represents more than one maturity range in order to diversify and minimize the effect of a change or twist in the term structure of interest rates. Symbolically, the returngenerating model can be represented by a multi-index market model like the one purposed by Lynge and Zumwalt [8], i.e., R,,, = A, + B, d,,,,, + C,%., + Dj%., + &,,
(2)
where &
is the return
R,,, is the return
on a short-term
bond,
on a long-term
l?,,., is the unsystematic
equity
return
A,, B,, C,, and D, are security ER,,, = Aj + B, ER,,,
bond, on a security
j,
specific constants,
+ C, ER,,,
+ 0, ER ,.,,
and a; = B; a: + CT a: + D; o: + uzE. A disciple of a the Sharpe-Lintner (2) as a special case of (l), where
security market in the following
line would view market four conditions hold,
model
I) A, = a,, 2) Aj + BIER,.,
+ C,ER,,,
+ D,ER,,,
= aj + biER,,,,
3) B; u,f + C: at + Df a: = by ai, and 4) 0: = a:. Based on the findings of the previous research concerning the interest rate sensitivity of equity prices [6, 7, 8, and 111, one can conclude that stock prices are interest rate sensitive. It is the primary concern of this study to determine if the implicit treatment of a portfolio’s interest rate sensitivity in the return-generating model, eq. (1) is a source of systematic bias in portfolio return characterizations. If a portfolio’s interest rate sensitivity is independent on the portfolio’s systematic risk and leverage, then the implicit treatment of the interest rate sensitivity in a portfolio-return-generating model such as model (1) does not bias portfolio return characterizations. Using a time series test for the independence of two random variates, (see Appendix A) the independence, or lack thereof, between both a short-term and long-term interest-rate index and each of ten portfolio price indexes will be examined to determine if the interest rate sensitivity of a portfolio price index is systematically related to the portfolio’s systematic risk or leverage. The data set used in this study will be described in the next section. Construction
of Stock
Price
Indices
and
Interest
Rate
Series
The stock price indices were formed from firms that are included on both the CRSP and COMPUSTAT data bases; they have December fiscal year ends and complete
88
.I BUSN
RES 1987:15:85-92
L.J. Johnson et al.
data over the interval l/74 to 12/78. This screening process netted 726 firms. These firms were then divided into five portfolios based on median debt ratios over the period. Each of these five portfolios were then divided into two portfolios based on the betas of their included firms. Each of the resulting ten portfolios then contained 72 or 73 securities. A stock price index was constructed for each of the ten portfolios, assuming that the firms included in the index remained constant throughout the period. Each index was equally weighted, with rebalancing occurring at the end of each month. The data source for the two interest rate time series is the H-15 release of the Federal Reserve. This release is published weekly, and contains interest rate observations for each trading day of the prior week. The observations were taken from the last trading day of each month. The next section reports the time series statistics i,,(z,n), ?,lk(~,n), SZ(n,q), and Sz(n,k) as described in Appendix A.
Conclusions
and
Implications
The major conclusion of this study deals with the hypothesis developed earlier. This paper demonstrates that Sharpe’s systematic risk measure may be a biased measure of portfolio returns with respect to interest rate sensitivity. This inference can be drawn by examining the test results. Note that the portfolio’s interest rate sensitivity, or lack thereof, follows a pattern with respect to p (6,) but not with respect to the leverage employed by the firms in the portfolio. The interest rate sensitivity of portfolio prices with respect to the short-term interest rate was found to be similar for both high and low systematic risk portfolios. However, the highrisk portfolio price’s have been shown to be independent of the long-term interest rate while the low-risk portfolio price’s are sensitive to the long-term interest rate. In general, the interest rate sensitivity of low-risk portfolios is greater than that of high-risk portfolios because the latter are sensitive to both the short-term and long-term interest rates, whereas the high-risk portfolios are sensitive to only the short-term interest rate. This implies that a return-generating market model should include more than just an equity market index. In addition it should include a debt index that represents a range of maturities or a set of debt indices that represent different maturities, i.e.,
R,,,= A, + B,&, +
C$,,,
+
D,&
+ E,,,.
Noting the caveat concerning the interpretation of the i,,, (z,n) and i,, (z,n) presented in Appendix A, if we assume that the cross correlations are normally distributed with standard errors equal to [1.96(T - I z I) “‘I, there is evidence that interest rates lead stock prices (see Table 1 and Table 2). It is clear from the significance of coefficients for a lead of one, as well as the sign distributions over the lag/lead range and trends in coefficients, that there is evidence that interest rates tend to lead stock prices. The most significant coefficients are at a lead of one period for three-month bill rate changes for both low- and high-beta portfolios. For long-term bond rate changes there is weaker evidence of significance at a leadof-one period for both low- and high-beta portfolios. This suggests that a next step in analyzing the interest rate sensitivity of equity prices should be along the lines of a “causality” or “lead/lag” time series test.
Interest-Rate
J1987:15:X5-92 BUSN RES
Sensitivity
Table 1. Cross Correlations: January
1974 to December
Three-Month 1978
Bill Rate Changes
89
and Stock Price Changes-
Low Beta Portfolios
Lag
Low Debt
Med-lo Debt
Medium Debt
Med-hi Debt
High Debt
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
0.05 0.16 0.23 -0.08 0.29 0.04 -0.04 -0.40 -0.02 -0.08 -0.14 -0.18 0.12
0.11 0.14 0.17 -0.03 0.17 0.03 -0.09 -0.41” 0.09 -0.05 -0.17 -0.17 0.23
0.16 0.15 0.08 0.03 0.13 -0.00 -0.13 -0.48 0.07 -0.04 -0.25 -0.14 0.23
0.10 0.11 0.16 -0.07 0.05 0.06 -0.22 -0.57” 0.12 -0.07 -0.23 -0.09 0.16
0.24 0.10 -0.01 0.13 0.13 0.04 -0.23 -0.35” 0.03 -0.03 -0.24 -0.13 0.18
24.95
24.39
28.19
32.76”
23.76
S,=
d.f. = 13
High Beta Portfolios
Lag
Low Debt
Med-lo Debt
Medium Debt
Med-hi Debt
High Debt
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
0.25 0.08 0.14 0.03 0.25 0.00 -0.22 -0.40 0.10 -0.18 -0.14 -0.05 0.10
0.08 0.20 0.18 0.14 0.10 0.12 -0.29 -0.32 0.02 -0.08 -0.26 -0.02 0.16
0.17 0.24 0.09 0.12 0.16 0.11 -0.28 -0.31” 0.02 0.03 -0.27 -0.05 0.16
0.18 0.04 0.20 0.08 0.18 0.03 -0.18 -0.41” -0.00 -0.07 -0.25 -0.03 0.11
0.12 0.16 0.28 0.05 0.22 0.06 -0.18 -0.36” -0.06 0.06 -0.27 0.02 0.13
s;, =
26.98
24.94
26.92
24.12
26.20
d.f. = 13
“Indicatessignificanceat 0.05 level.
References 1. Beaver, William. The Informational Content of Annual Earnings Empirical Research in Accounting: Selected Studies. 1968 Supplement Journal of Accounting Research (1968). Box, G. E. P. and Jenkins, G. J. Time Series Analysis: Francisco: Holden-Day, Inc., (1970).
Forecasting
Announcements. to Vol. 6 of the and Control.
San
Box, G. E. P. and Pierce, D. A. Distribution of Residual Auto-Correlations in Autoregressive-Integrated-Moving Average Time Series Models. Journal of the American Statistical Association. 65: 1509-26 (December 1970). Brown, Philip and Ball, Ray. An Empirical Evaluation Journal of Accounting Research. 6: 159-78 (Autumn Haugh, Larry D. Checking
the Independence
of Accounting 1968).
of Two Covariance
Income
Stationary
Numbers.
Time Series:
90
BUSN RES 1987:15:85-92
J
L.J. Johnson
Table 2. Cross Correlations: Long-Term Government Changes-January 1974 to December 1978
Rate Changes
et al.
and Stock Price
A. Low Beta Portfolios Low Debt
Med.10 Debt
Medium Debt
Med-hi Debt
High Debt
0.01 0.28 -0.01 0.05 0.06 0.13 -0.23 -0.30” 0.13 -0.07 -0.13 -0.19 -0.04 19.84
0.03 0.19 -0.08 0.11 0.02 0.16 -0.38” -0.16 0.29” -0.11 -0.15 -0.08 -0.02 22.40
0.05 0.14 -0.15 0.05 0.04 0.14 -0.41” -0.19 0.26 -0.10 -0.19 0.10 0.03 24.08”
0.18 0.08 -0.14 0.04 0.05 0.12 -0.44” -0.37” 0.22 -0.11 -0.13 -0.01 -0.07 28.76”
0.18 0.0s -0.19 0.05 0.15 0.03 -0.40 -0.28” 0.14 -0.06 -0.11 -0.14 0.08 23.97”
Lag -6 -5 -4 -3 -2 -1 0 1 2 3 4
5 6 s,=
d.f. = 13
B. High Beta Portfolios
Lag
Low Debt
Med-lo Debt
Medium Debt
Med-hi Debt
High Debt
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 s,=
0.12 0.10 -0.18 0.17 0.09 0.12 -0.31” -0.15 0.12 -0.11 -0.08 -0.11 -0.04 16.33
0.09 0.06 -0.09 0.17 0.11 0.18 -0.34 -0.18 0.16 -0.15 -0.20 -0.06 -0.05 19.95
0.10 0.07 -0.10 0.04 0.16 0.21 -0.38 -0.20 0.25 -0.13 -0.14 -0.09 -0.05 23.44”
0.10 0.05 -0.09 0.06 0.07 0.15 -0.30” -0.28” 0.19 -0.15 -0.13 -0.10 -0.05 18.41
0.17 0.07 -0.05 0.08 0.12 0.12 -0.28” -0.27” 0.15 -0.07 -0.18 -0.08 0.05 17.56
d.f. = 13
“Indicates significanceat 0.05 level
A Univariate Association.
Residual 71: 378-385
Cross-Correlation (June 1976).
Approach.
Journal
of the American
Statistical
6. Joehnk, Michael D. and Nielson, James F. The Effect of Interest Rates on Utility Share Prices. Review of Business & Economic Research. 12: 35-43 (Winter, 1976-77). 7. Lloyd, William P. and Shick, Richard Journal
of Financial
and Quantitative
A. A Test of Stone’s Two-Index Model of Returns. Analysis. 12: 363-76 (September 1977).
8. Lynge, Morgan J., Jr. and Zumwalt, J. Kenton An Empirical Study of the Interest Rate Sensitivity of Commercial Bank Returns: A Multi-Index Approach. Journal of Financial and Quantitative Analysis. 15: 731-42 (September 1980). 9. Malkiel, American
10. Ofer,
Burton
J. and Cragg, John G. Expectations Review. 60: 601-17 (September
Economic
Aharon
R. Investors’
Expectations
and the Structure 1970).
of Earnings
Growth:
Their
of Share Prices. Accuracy
and
Interest-Rate
J BUSN RES 1987:15:85-92
Sensitivity
Effects on the Structure of Realized Rates of Return. Journal of Finance. 30: 508-23 (May 1975). 11. Stone, Bernell K. Systematic Interest Rate Risk in a Two-Index Model of Returns. Journal of Financial and Quantitative Analysis. 9: 709-25 (November 1974).
Appendix
A
A Test for the Independence
of Two Time Series
The method used in this study to investigate the stochastic relationship between two covariance-stationary time series involves two stages. The first stage calls for fitting univariate models to each of the series Stage two entails cross-correlating the resulting two strictly stationary residual series. Haugh [5] has shown that under the null hypothesis of series independence, when the set of residual cross correlations are squared and then weighted and summed, the asymptotic distribution of this weighted average is x-square. The univariate dynamic model used in this study to describe the time series, p,,, (price index of the stock portfolios), and r,,, and r,,! (the short-term and long-term interest rate) is a nonlinear, rational-distributed lag model. Statistically, this model has been described by Box and Jenkins [2]. To fit such a univariate model Box and Jenkins suggest the three-step process of 1) specification of the models form, 2) estimation of the parameters of this model by a nonlinear, least-squares, maximum-likelihood method and 3) adequacy testing to determine the models fit. Assume that @,,n: t = 1,2, . . . $0 and n = 1,2, . . ,lO}, {T,,~:r = 1,2, . . . ,60} and by a mixed auto-regressive{r,.j t = 1,2, . . . ,60} can be modeled individually integrated-moving average model, i.e.,
fn(B)(l - qp,,, = g,(wL,,, n = 12,. . . 710,
(4)
and i(B)(l v(B)(l
- B)r,,, - Q,.,
= j(B)&, = m(B)&
(5)
normal series. Using the specification and where - denotes a strictly stationary estimation technique provided by Box and Jenkins, the parameters (f,(B), g,(B), (these results will be provided upon request). i(B), v(B), m(B)] were estimated The Box-Pierce [3] test statistic was used to examine the adequacy of the fitted models (Only two of the fitted models were not strictly random-walk or firstdifference models. The low-risk and low debt-to-equity price index and r,,, demonstrated the need for a low-order moving average component that was removed, resulting in a strictly stationary, residual-time series). The next stage in testing for the independence of two time series is to calculate the cross correlations between the residuals from the univariate models. Specifically, the cross correlations of the ten portfolio residuals with respect to both the short-term and long-term interest-rate residuals were calculated as follows, i.e.,
92
J BUSN RES 1987:15:RS-92
L.J. Johnson
et al.
where n = 1,2, . . . ,lO; z = 6,5, . . . , - 5, -6; and - denotes that these variables are the residual estimates of the true white-noise sequences (These residual cross correlations, fh,,(z,n) and ?,&,n) are reported in Tables 1 and 2). It is important to note that the individual i,,(z,n) and ?,,,,(z,n) can’t be examined for significance since the true parameter values are not known and hence the covariance matrix of the individual cross correlations are unknown. However, Haugh [5] has shown that the following test statistic, S$ is approximately distributed as x-square with 2M + 1 degrees of freedom under the assumption of series independence, i.e., S;(q,n)
= T *
c
(T - I z
I)-‘f&(z,rz),
*=-m
(7)
$,,(/+I) = T * 2 (T - I z I)-‘%(z>& where T = 60 and m = 6. These results can be used to test the independence of two covariance-stationary series (The Sz(q,n) and Sz(k,n) are reported in Tables 1 and 2).