Previous studies on the
magnitude of the market risk premium have tended to focus on the historical
average difference between some proxy for the market portfolio and the yield on
the Treasury bill. The size of the risk premium has exhibited a strong
dependency on the time interval over which the return was based. Jagannathan,
McGrattan and Scherbina (2001) finds that the US equity premium averaged 8.9
percent during
the fifties and 3.98 percent in the nineties. Siegel (1992), based on the arithmetic
mean of the short-term risk-free rate and the real return on equity, reported an
equity risk premium of 1.99 percent and 6.92 percent for the periods 1800-1888 and 1889-1978,
respectively. For the period 1928-1998, Siegel (1998,1999) report an equity
risk premium over the Treasury bill of 8.6 percent. Dimson, Marsh, and Staunton (2002)
computed the equity risk premium for 92 overlapping decades across 16 countries
of 5.1 percent.
Arnott and Bernstein (2002),
argue that basing the magnitude of the equity risk premium on historical
realized return is misleading because the “ the market exceeded objective
expectations as a consequence of a series of historical accidents”. These
accidents they list as follows: 1) decoupling of yields from real yields, 2)
rising of valuation multiples, 3) survivor bias, and 4) regulatory reforms. They argue that the historical return as measured by the market index ignores
the fact that “An important engine for economic growth is the creation of new
enterprises. The investor in today’s enterprises does not own tomorrow’s new
enterprises- not without making a separate investment in those new enterprises
with new investment capital”. They demonstrate that “the historical average
equity risk premium, measured relative to 10-year government bonds as the risk
premium investors might objectively have expected on their equity investments,
is about 2.4 percent”.
Establishing the long-run
relationship between a pair of time series requires that the variables be
cointegrated. If a series must be differenced d times before it becomes
stationary, it is said to contain d unit roots and integrated of order
d, expressed as I(d). For two series yt and xt
to be cointergrated, they each must be integrated of the same order d.
Moreover, there must exist a vector β, such that the error term from the
regression (εt =yt – βxt) is of a lower
order of integration, I(d-b), where b>0. According to Engle and
Granger (1987), the two series are said to be cointegrated of order (d, b). Cointegration implies that even though the two or more series themselves may
contain stochastic trends, the series are linked to form an equilibrium
relationship to which the system converges over time. The error term, εt,
can be interpreted as the distance that the system is away from equilibrium at
time t.
In order to investigate the long-run relationship between market risk premium
and the risk-free rate, it is necessary to determine the integration of these
variables. To achieve this, the augmented Dickey-Fuller (ADF) test of Said and
Dickey (1984) and Said (1991) is employed. The test is based on the null
hypothesis of unit root with drift process against the alternate of trend
stationary process. The selection of the optimal lag length is based on the Akaike Information criterion (AC), multivariate Hannan-Quinn (1979) criterion
(HQ), and multivariate Schwarz (1978) Bayesian (SC) criterion.
Given the integration characteristics of the market risk premium and risk-free
rate variables, the Johansen maximum likelihood procedure is used to test for
possible cointegration between the variables. Johansen (1988; 1991) and
Johansen and Juselius (1990) procedure is based on the error-correction
representation of the vector autoregression VAR(k) model with Gaussian
errors. Johansen and Juselius (1990) likelihood-ratio tests (Lambda-max and
Trace tests) are used to determine the number of cointegrating vectors based on
the maximum likelihood estimates of the cointegrating vectors. The lag length,
k, is chosen by a combination of AC, HQ, and SC criteria.
Granger (1986, 1988) pointed out that if two variables are cointegrated, then
Granger causality must exist in at least one direction. Granger (1969)
describes a variable xt as Granger causing another variable yt,
if the inclusion of lagged values of x improves the forecast of y. Engle and
Granger (1987) point out that the standard Granger causality tests are
inappropriate and misleading in the presence of cointegration. Standard Granger
causality tests that are augmented with error-correction terms, obtained from
the cointegrating relationship, are used to investigate the long-run effects. According to Engle and Granger (1987), such tests assure that valid inferences
can be made on variables that are cointegrated.
STATIONARITY TEST RESULTS
The augmented
Dickey-Fuller (ADF) test was conducted using the following regression model:
| Δzt
= α0 + βzt-1 + ηT + |
 |
1 |
Where zt is the time series, T is a time trend, and μt is
white noise. Two versions of the ADF test are conducted. The first version
tests the null that at is a non-stationary
process (h is
set equal to zero) and b
= 0 against the alternate that z(t) is a stationary process,
b<0. In the second
version, the null hypothesis is that the time series zt is a unit
root with drift process: β = 0, against the alternate that zt
is a trend stationary process: β <0. The test statistic is the t-value of β.
The selection of the optimal p was based on the Akaike, Hannan-Quinn and
Schwarz information criteria. The results, though not presented in the table,
are available upon request. The results of the stationarity or unit root tests
are presented below in Table 1.
|
TABLE 1
Stationarity Test Results |
|
PANEL A: Variables in Level |
|
|
Test
#1:
H0:
Unit Root
H1:
Stationary Process |
Test
#2
H0:
Unit Root with Drift
H1:
Linear Trend Stationary |
|
Variables |
t-Statistic |
t-Statistic |
|
Ten-Year |
-1.1847 |
-0.8831 |
|
One
Year |
-1.6687 |
-1.0147 |
|
Three-Month |
-1.6586 |
-1.5498 |
|
NYSETen |
-5.3581** |
-5.8289** |
|
NYSEOne |
-3.2984** |
-3.6054** |
|
NYSEThree |
-5.3068** |
-5.7838** |
|
SPTen |
-5.0062** |
-5.1671** |
|
Spone |
-3.1534** |
-3.2963* |
|
SPThree |
-4.9781** |
-5.1512** |
|
PANEL B: Variables in First Difference |
|
TenYear |
-5.9294** |
-5.9759** |
|
OneYear |
-6.1830** |
-6.3401** |
|
ThreeMonth |
-6.3017** |
-6.3539** |
|
|
Critical values:
5
percent :
-2.89, 10 percent: -2.58 |
Critical values:
5 percent: -3.40, 10 percent: -3.13 |
|
|
* significant at the 10 percent level
** significant at the 5 percent level |
The test results indicate that all the measures of the risk-free interest rate
are non-stationary in their levels and stationary in the first difference. In
contrast, all the measures of the market risk premium are stationary in their
levels at the 5 percent level of significance with the exception of Spone, which is
linear trend stationary at the 10 percent level. Since the risk premium variables are
stationary, testing for the stationarity of the first difference is not
necessary. Cointegration requires that the variables be integrated of the same
order. Since the risk free rate measures are integrated of order one, I(1),
and the risk premium measures are integrated of order zero, I(0), the
cointegration analysis can not be performed using the levels of the variables. The lack of cointegration implies that there is no long-run relationship between
the levels of the variable. This does not necessarily preclude the existence of
a short-run effect between the risk-free rate and the risk premium. To
investigate the existence of this relationship, an ordinary least squares
regression (OLS) is performed for each pair of risk premium and risk free rate
at their stationary form.
ORDINARY
LEAST SQUARES REGRESSION RESULTS
The estimated
regression model is of the form:
Yt
= a0
+ b0
Xt + b1L1
+b2L2
+ b3L3
+ b4L4
+ b5L5
+ b6L6
+ ut
2
where: Yt
is the market risk premium for the corresponding risk free rate
Xt,
is the first difference of the risk-free rate and
L1
through L6 are the lagged value of Xt.
Initial estimation of the model show significant autocorrelation and
hetroskedasticity based on the Durbin-Watson and Breusch-Pagan tests,
respectively, for the models based on the three month T-bill and the ten-year
T-note. For the one- year T-note based model, the null of homoskedasticity
could not be rejected at the 10 percent level of significance. The models are
re-estimated in the form of OLS with ARMA errors (Bierens, 2003). An error
specification of the form:
[1
– a(1,1)L]ut = et 3
with et white noise and L the lag operator is
assumed. The results are presented below in Table 2.
|
TABLE 2
Results of OLS Regression with ARMA Errors
(t-values in parenthesis) |
|
Risk-Free Rates |
Three Month T-Bill |
One Year T-Note |
Ten Year T-Note |
|
Market Risk Premium |
NYSE |
S&P 500 |
NYSE |
S&P 500 |
NYSE |
S&P 500 |
|
X1 |
-2.5813
(-3.958)*** |
-2.563
(-3.987)*** |
-4.404
(-9.136)*** |
-4.085
(-8.64)*** |
-6.543
(-6.127)*** |
-6.162
(-5.84)*** |
|
L1 |
-2.0075
(-2.556)** |
-2.569
(-3.315)*** |
-3.847
(-6.845)*** |
-3.68
(-6.672)*** |
-5.106
(-3.907)*** |
-4.929
(-3.814)*** |
|
L2 |
-2726
(-3.172)*** |
-3.137
(-3.694)*** |
-3.483
(5.577)*** |
-3.427
(-5.589)*** |
-5.903
(-4.062)*** |
-5.683
(-3.953)*** |
|
L3 |
-3.861
(-4.395)*** |
-4.213
(-4.852)*** |
-3.063
(-4.813)*** |
-2.99
(-4.783)*** |
-5.137
(-3.389)*** |
-5.455
(-3.628)*** |
|
L4 |
-3.405
(-3.955)*** |
-2.801
(-3.292)*** |
-2.716
(-4.354)*** |
-2.748
(-4.485)*** |
-4.689
(-3.229)*** |
-5.042
(-3.516)*** |
|
L5 |
-2.014
(-2.566)** |
-2.032
(-2.621)*** |
-1.632
(-2.91)*** |
-1.639
(-2.976)*** |
-1.637
(-1.255) |
-2.153
(-1.668)* |
|
L6 |
-0691
(-1.059) |
-0.603
(-0.938) |
-0.782
(-1.626) |
-0.855
(-1.811)* |
-0.938
(-0.877) |
-1.135
(-1.074) |
|
a0 |
0.963
(0.272) |
-4.968
(-1.175) |
4.119
(1.434) |
-0.580
(-0.195) |
0.551
(0.159) |
-4.072
(-1.113) |
|
a(1,1) |
0.919
(65.0)*** |
0.934
(66.37)*** |
0.925
(64.37)*** |
0.929
(64.86)*** |
0.918
(63.77)*** |
0.924
(64.04)*** |
|
R-Square |
0.8333 |
0.8313 |
0.862 |
0.861 |
0.8313 |
0.8298 |
|
N |
860 |
860 |
729 |
729 |
860 |
860 |
|
***, **,
and * indicate significance at the 1 percent, 5 percent, and 10
percent levels,
respectively. |
The results of the OLS
regression estimation show that changes in the risk free rate as proxied by the
three- month treasury bill, one-year and ten-year Treasury notes have a
significant effect on the market risk premium. These effects are significant at
the 1 percent level up to four months after the change occurs. The model as whole,
with the ARMA error term, explains more than 80 percent of the risk premium based on
the coefficient of determination. In the absence of the ARMA error term, the
explanatory power of the risk-free rate varied with the proxy for the risk-free
rate. The three-month Treasury bill explained 2.98 percent and 2.85 percent of the level of
the risk premium based on the NYSE, and S&P 500 index, respectively. These
values are 4.1 percent and 4.04 percent for the ten-year Treasury note and 6.58
percent and 6.38 percent for
the one–year treasury note.
CONCLUSION
This study looked at the relationship between market risk premium and the risk
free rate by constructing a running annual return on a market portfolio and
using the monthly yield on proxies for the risk-free rate, to determine the risk
premium. This approach, in contrast with the calendar annual return commonly
used in the literature, increases the power of the results through a larger
sample size. This is significant since Treasury securities are auctioned on a
monthly basis. The stationarity properties of the resulting time series were
studied as well as the relationship between the stationary time series. The
results suggest that the market risk premium is a level stationary process,
whereas the measures of the risk premium are first difference stationary. Consequently a bivariate long-run relationship cannot exist between the
risk-free rate and the market risk premium. An OLS regression with ARMA errors
analysis indicates a strong explanatory power of the changes in the risk-free
rate on the market risk premium. There exist a strong explanatory power in the
error term, based on the jump from a coefficient of determination value less
than 7 percent, without the error term, to greater than 80 percent with its inclusion. This
indicates that the risk premium is largely determined by other variables that
are being proxied by the error term. The stationarity of the risk premium in the
level suggests that the risk premium series has no memory. This lends support
to the use of long-run averages as a measure or forecast of the market risk
premium. The significance of the error term suggests that further research is
needed to determine the variables that are embedded in the error term. This
knowledge will resolve the problem of possible negative equity risk premium and
help provide an objective measure and forecast of the market risk premium.
REFERENCES
Arnott, Robert D., and Peter L.
Bernstein (2002) “What Risk Premium is “Normal”?” Financial Analyst Journal,
58, 2, 64-85.
Bierens, H. J. (2003)
“EasyReg International”, Department of Economics, Pennsylvania State University,
University Park, PA.
Blanchard, O.J. (1993) “Movements in the
Equity Premium,” Brookings Papers on Economic Activity, 2, 75-118.
Dimson, E., P. Marsh, and M. Stuanton,
(2002) “Triumph of the Optimist: 101 Years of Global Investments Returns,
Princeton, NJ, Princeton University Press.
Engle, Robert F., and C.W.J.
Granger,(1987) “Co-integration and Error Correction: Representation, Estimation,
and Testing.” Econometrica, 55, 251-76.
Granger, C.W. J.,(1969) “Investigating
Causal Relations by Econometric Models and Cross Spectral Methods.” Econometrica, 424-438.
---------(1986) “Developments in the
Study of Cointegrated Economic Variables.” Oxford Bulletin of Economics and
Statistics, 48, 213-228.
---------(1988) “Some Recent
Developments in a Concept of Causality.” Journal of Econometrics, 39,
199-211.
Hannan, E.J. and B.G. Quinn (1979) “The
Determination of the Order of an Autoregressive.” Journal of the Royal
Statistical Society, January, 190-195.
Jagannathan, R., E. R. McGrattan, and A.
Scherbina (2000) “The declining US Equity Premium,” QuarterlyReview,
Federal Reserve Bank of Minneapolis, 24, 4, 3-19.
Johansen, S., (1988) “Statistical
Analysis of Cointegration Vectors,” Journal of Economic Dynamics and Control,
12, 231-254.
----------, (1991) “ Estimation and
Hypothesis Testing of Cointegration Vectors in Gaussian Vector Autoregressive
Models,” Oxford Bulletin of Economics and Statistics, 54, 1551-80.
Johansen, S. and Katarina Juselius
(1990) “ Maximum Likelihood Estimation and Inference on Cointegration with
Application to the Demand for Money.” Oxford Bulletin of Economics and
Statistics, 52, 169-210.
Kairys, Joseph P. Jr. (1993) “Predicting
Sign Changes in the equity Risk Premium Using Commercial Paper Rates,” Journal of Portfolio Management,
20, 1,41-51.
O’Hanlon, John and Anthony Steele (2000)
“Estimating the Equity Risk Premium Using Accounting Fundamentals,” Journal
of Business Finance & Accounting, 27, 9/10, 1051-1083.
Said, S.E.,(1991) “Unit Root Test for
Time Series Data with a Linear Time Trend.” Journal of Econometrics
47,
285-303.
Said, S.E., and D. A. Dickey (1984)
“Testing for Unit Roots in Autoregressive Moving Average of Unknown Order.” Biometrika,
71, 599-607.
Schwarz, G., (1978) “Estimating the
Dimensions of a Model.” Annals of Statistics, 461-464
Siegel, J.J. (1992) “The Real Rate of
Interest from 1800-1990: A study of the US and the UK,” Journal of Monetary
Economics 29, 227-252.
----------- (1998) “Stocks for the
Long Run, 2nd ed., New York, NY, McGraw-Hill.
---------- (1999) “The Shrinking Equity
Premium: Historical Facts and Future Forecasts,” Journal of Portfolio
Management 26, 10-17.
Tippet, Mark (2000) “Discussion of
Estimating the Equity Risk Premium Using Accounting Fundamentals,” Journal of
Business Finance & Accounting; Oxford27, 9/10,1085-1106.
Wadhwani, S.B., (1999) “The US Stock
Market and the Global Economic Crisis,” National Institute Economic Review,
86-105.
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Confidence
W. Amadi confidence.Amadi@famu@edu
is an Associate Professor of Finance at Florida A&M University.