Journal of Forecasting, Vol. 13, 335-367 (1994) Forecasting Stock Returns An Examination of Stock Market Trading in the Presence of Transaction Costs M. HASHEM PESARAN Trinity College, Cambridge ALLAN TIMMERMANN Birkbeck College, University of London and CEPR ABSTRACT The paper presents new evidence on the predictability of excess returns on common stocks for the Standard and Poor’s 500 and the Dow Jones Industrial portfolios at the monthly, quarterly, and annual frequencies. It shows that recursive predictions obtained on the basis of the excess returns regressions are capable of correctly predicting a statistically significant proportion of the signs of the actual returns. The paper also shows that the switching portfolios constructed on the basis of the signs of the recursive predictions mean-variance dominate the respective market portfolios when trading takes place on a quarterly or annual basis. This result holds even under a high transaction cost scenario. However, due to the larger number of transactions at the monthly frequency the monthly switching portfolios only mean-variance dominate the respective market portfolios when transaction costs are zero or low. KEY WORDS Forecasting stock returns Portfolio selection rules Transaction costs SP500 INTRODUCTION Over the past few years a number of studies have shown that a substantial proportion of monthly, quarterly, and annual variations in excess returns on common stocks can be predicted using ex ante dated regressors. Notable examples are the papers by Balvers et al. (1990), Breen et al. (1990), Campbell (1987), Cochrane (1991), Cutler et al. (1988), Fama and French (1988, 1989), French et al. (1987), Pesaran (1991), Poterba and Summers (1988) and Sentana and Wadhwani (1991). These studies are concerned with ‘predictable volatility’. Other papers focus on ‘unpredictable volatility’ and provide an explanation of the variations in excess returns in terms of contemporaneously dated and/or future dated regressors (see French, Schwert, and Stambaugh, 1987; Fama, 1990). In this paper we present new evidence on the predictability of excess returns and examine the possibility of forming an investment strategy, based on recursive predictions of excess returns, which is more profitable and less risky as compared to a buy-and-hold strategy in the market. CCC 0277-6693/94/040335-33 Received August 1992 0 1994 by John Wiley & Sons, Ltd. Revised August I993
Transcript
Journal of Forecasting, Vol. 13, 335-367 (1994)
Forecasting Stock Returns An Examination of Stock Market Trading in the Presence of Transaction Costs
M. HASHEM PESARAN Trinity College, Cambridge
ALLAN TIMMERMANN Birkbeck College, University of London and CEPR
ABSTRACT The paper presents new evidence on the predictability of excess returns on common stocks for the Standard and Poor’s 500 and the Dow Jones Industrial portfolios at the monthly, quarterly, and annual frequencies. It shows that recursive predictions obtained on the basis of the excess returns regressions are capable of correctly predicting a statistically significant proportion of the signs of the actual returns. The paper also shows that the switching portfolios constructed on the basis of the signs of the recursive predictions mean-variance dominate the respective market portfolios when trading takes place on a quarterly or annual basis. This result holds even under a high transaction cost scenario. However, due to the larger number of transactions at the monthly frequency the monthly switching portfolios only mean-variance dominate the respective market portfolios when transaction costs are zero or low.
Over the past few years a number of studies have shown that a substantial proportion of monthly, quarterly, and annual variations in excess returns on common stocks can be predicted using ex ante dated regressors. Notable examples are the papers by Balvers et al. (1990), Breen et al. (1990), Campbell (1987), Cochrane (1991), Cutler et al. (1988), Fama and French (1988, 1989), French et al. (1987), Pesaran (1991), Poterba and Summers (1988) and Sentana and Wadhwani (1991). These studies are concerned with ‘predictable volatility’. Other papers focus on ‘unpredictable volatility’ and provide an explanation of the variations in excess returns in terms of contemporaneously dated and/or future dated regressors (see French, Schwert, and Stambaugh, 1987; Fama, 1990). In this paper we present new evidence on the predictability of excess returns and examine the possibility of forming an investment strategy, based on recursive predictions of excess returns, which is more profitable and less risky as compared to a buy-and-hold strategy in the market. CCC 0277-6693/94/040335-33 Received August 1992 0 1994 by John Wiley & Sons, Ltd. Revised August I993
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Building on the findings in the literature we report regression results for the annual, quarterly, and monthly excess returns on Standard & Poor’s 500 (SP 500) and Dow Jones’ Industrial (DJ) portfolios. We find that dividend yields, the rate of inflation, the change in industrial production, and various measures of interest rates have substantial predictive power for excess returns. These regressions are used to form recursive and ex ante predictions of excess returns. The recursive predictions are utilized to construct a simple trading rule whereby the investor holds stocks if the ex ante prediction of excess returns is positive, otherwise holds bonds. The performance of this ‘switching portfolio’ is then compared with the market portfolio. Our results show that it is more difficult to make ex ante predictions of excess returns during the stable 1960s compared to the more volatile 1970s and 1980s. This finding suggests that the extent to which excess returns are predictable may depend on the size of the shocks that the economy is subject to and the speed with which financial markets adjust to these shocks. The large shocks to the economy during the 1970s and the 1980s by creating new uncertainties in the financial markets as well as changing the corporate earning prospects have also substantially altered the market’s perception and anticipation of risks.
The ability to predict stock returns does not necessarily imply that it will be possible to exploit the predictions for the purpose of making money. Important classes of models used in forecasting imply that the predicted risk premia are always positive. This holds for GARCH- in-mean models as well as regression equations that only include the dividend yield as a regressor (Shiller, 1984). To perform better than holding the market portfolio it is necessary that the forecasting equation is capable of accurately predicting negative excess returns at least for some periods. Otherwise it will not be possible to generate profits in excess of the return on the market portfolio.
Trading in bonds and stocks entails transaction costs that cannot be neglected. Such costs generate an important trade-off in the timing of investment decisions. When an investor has market-timing skills, and there are no transaction costs, the optimal investment strategy is often to trade as frequently as possible. Frequent trading has an obvious advantage since the investor profits from predicting a larger number of times where the market turns. The paper sheds light on the issue of the optimal trading frequency in the presence of transaction costs, and provides a comparison of trading strategies based on annual, quarterly, and monthly frequencies.
At the annual and quarterly frequencies our results show that the switching portfolios pay a higher mean return than the market portfolio over the period 1960-90. This conclusion holds even under a high transaction cost scenario where it is assumed that transaction costs for trading in stocks and bonds are 1.0% and 0.1% of the sum invested, respectively. However, the frequent switching between stocks and bonds at the monthly trading horizon implies that the mean returns on the monthly switching portfolios are smaller than mean returns on the market indices when transaction costs are high. For example, in the case of the SP 500 portfolio, under the zero transaction cost scenario the differences in mean returns (computed at an annual rate) between the market portfolios and the switching portfolios are 1.9% for trading at annual frequencies, 2.2% for trading at quarterly frequencies, and 2.3% for trading at monthly frequencies. The difference between the mean returns of the two portfolios decreases when transaction costs are incorporated in to the analysis. In the case of the high transaction cost scenario the differences between the mean returns on the switching and market portfolio decreases to 1.5%, 1.1%, and -1.0% for the annual, quarterly, and monthly frequencies, respectively. Similar results were obtained for the Dow Jones Portfolio.
On the issue of comparing the trading strategies for the SP 500 index across frequencies, mean returns from the quarterly and monthly switching portfolios were higher than the mean
M. Hashem Pesaran and Allan Timmermann Forecasting Stock Returns 331
return from the annual switching portfolios in the zero transaction cost scenario. In the high transaction cost scenario the mean return on the annual switching portfolio was higher than the mean returns on the monthly and quarterly switching portfolios. The annual switching portfolio also has a lower standard deviation than the quarterly or monthly switching portfolios. These results indicate that trading more frequently than at the annual horizon only pays off if cost transaction costs can be kept at a very low level (less than 0.5%).
The paper also applies a new predictive failure test developed by the authors (Pesaran and Timmermann, 1992) to the recursive predictions of excess returns. This test focuses on the ex ante prediction of the sign of the excess return variable and has the virtue of being distribu- tion-free. At all frequencies the test results are statistically significant, providing additional evidence on the market timing performance of the excess return forecasting equations.
The outline of the paper is as follows. The next section reports the annual, quarterly, and monthly regression results. The third section examines the statistical significance of the recursive prediction of excess returns. The fourth section evaluates the economic significance of a trading strategy based on the recursive predictions. The fifth section offers some conclusions.
The paper ends with a postscript where the preferred excess return regressions are re- estimated using the additional annual, quarterly, and monthly observations for the years 1991 and 1992 that have become available since the first version of this paper was completed and submitted for publication. This is done in response to the likely criticism that the excess return predictions reported in the main body of the paper are not truly ex ante, as they are based on regression equations derived from estimates over the whole sample period (1954-90). * The results reported in the postscript are the nearest that we could come in producing truly ex ante forecasts. These additional forecasts turn out not to be very helpful in shedding any further light on the relative performance of the switching portfolio as compared to a buy-and-hold strategy. Almost all the forecasts of excess returns over the 1991-92 period turned out to be positive, thus rendering the switching portfolio identical to the market portfolio. Passage of time is now needed before further evidence on the predictive performance of the excess return regressions in this paper can be obtained.
EXCESS RETURN REGRESSIONS
The literature on the predictability of returns or excess returns on common stocks is by now quite extensive. It has been shown that a substantial part of variations in excess returns is predictable. Campbell (1987) finds that 1 1 % of the monthly variations of excess returns on the CRSP portfolio can be explained by a 1-month T-bill rate, a measure of the term premium and its lagged value over the period 1959-79. Fama and Freneh (1989) show that for the period 1953-87, 17% of annual variations in excess returns on the value weighted CRSP portfolio can be explained by using past dividend yields and a measure of the term premium as regressors. However, they obtain substantially weaker results at a quarterly frequency, where they manage to explain only 8% of the variations in excess returns over the period 1953-87. Pesaran (1991) explains over 60% of the variations in the annual SP 5 0 0 excess returns over the period 1948-86 by using lagged changes in the commercial paper rate in addition to lagged values of the inflation rate and the dividend yield as regressors.
' This exercise was suggested to us by Sean Holly, an editor of this journal.
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Let Pt be the nominal closing stock price on the last trading day of January in year t , and denote the nominal dividends paid during year t - 1 by Dt- 1 *. Then the annual rate of return on the SP 500 index, Rr, is given by3
(1) Rt = ( f‘t + Dt - I - Pt - I ) /Pt - I
Rr = (Pt + Dt - Pt - 1 )/PI- 1
For the quarterly and monthly data returns are defined as
(1 ’1 where Pt is the closing stock price on the last trading day of the quarter (month) and Dt is dividends paid during the quarter (month). Excess returns, pt, are given by
pt = Rt - i t - l (2)
where it- 1 represents the T-bill rate at the end of the month in which Pt- 1 is measured. For the annual trading scheme we restrict the information set, Or, to contain observations
that are publicly available on the last trading day of January. In particular, we assume Ot includes observations on financial statistics that are made available daily such as share prices and interest rates, and monthly data such as the Producer Price Index (PPI), which is typically published with a lag of 15-20 days. This implies, for instance, that Or can contain data on interest rates and the stock price index on the last trading day in January of the trading year, whereas the Producer Price Index (PPI) has to be dated in December of the previous year or earlier.4 At the quarterly and monthly frequencies financial data such as stock prices and interest rates are measured with a lag of 1 month while macroeconomic data and producer prices are only available with a two-month lag.
As far as the choice of the regressors in the excess return equations is concerned, we focused on the variables considered to be important in the literature. The most important of these variables are those that, directly or indirectly, are thought to have an influence on the equity risk premium. Rozeff (1984), for example, argues that changes in the dividend yield is an important proxy for the variations in the risk premium. Other suggested proxies for the equity risk premium are the default premium on bonds or volatility-based measures obtained from past variations in stock returns (French et al., 1987).
Fama and French (1989) argue that the propensity to save out of wealth varies pro-cyclically such that the required rate of return on assets must vary counter-cyclically for equilibrium to obtain. In this wide sense, variables that trace the state of the economy might be related to the return on stocks. This suggests the use of interest rates that are known to follow a pattern related to the business cycle.
Balvers et al. (1990) derive an equilibrium model in which expected stock returns is a function of aggregate output, and their empirical analysis using annual data over the period
Note that this introduces a minor error in the measurement of annual dividends, since the dividend data are measured from the beginning of January to the end of December of the previous year, rather than from the end of January to the end of January of the previous year as required here. However, given the fact that most of the variations in excess returns originate from price variations this error is not likely to have any significant effect on the results. ’These and other variables used in this study and the associated data sources are described in the Data Appendix. Note that in the case of annual series the returns are ‘backward dated’, in the sense that the rate of return, say for 1987, is measured from January 1986 to January 1987. 4The excess returns regression on the SP 500 portfolio reported in Pesaran (1991, Table 10) used the lagged value of real dividend yields, the change in the nominal 6-month commercial paper rate for July and an inflation rate as regressors. The purpose of that study was to explain the variation in stock returns, possibly by using contemporaneously dated regressors. However, for the purpose of prediction, these variables are not all contained in the public information set at the end of January when the trade is to take place.
M. Hashem Pesaran and Allan Timmermann Forecasting Stock Returns 339
1947-87 suggests that industrial production (as a proxy for aggregate output) has predictive power for stock returns. This is in line with the work by Chen et al. (1986), which uses growth rates in industrial production in addition to a number of interest rates as factors in their cross- sectional return equations. Finally, inflation effects (sometimes decomposed into unexpected and expected components) on stock returns have been used extensively in the literature.
On the basis of the results in the literature we carried out a number of regressions using various measures of interest rates, inflation rates, the dividend yield, and the change in the industrial production index. We report these first. Subsequently we report further regression results on the statistical significance of business cycle indicators and non-linear effects for the prediction of excess returns.
Annual excess returns regressions We experimented with a number of specifications for the annual excess returns on the SP 500 portfolio over the period from January 1954 to January 1991. This sample period was chosen on grounds of data availability (the 12-month T-bill rate series starts in June 1952). Focusing on lagged observations on dividend yields, inflation, interest rate changes, and a term premium, measured as the difference between the 6-month commercial paper rate and the 3-month T-bill rate in January, we managed to explain around 60% of the total variation in the excess return over the 1954-91 period. The regression result together with the associated test statistics are reported in Table I. The definition of the variables, the exact time of their measurement, and the sources of the data used are given in the Data Appendix.
Consider first the effect of the lagged inflation rate on excess returns. The market seems to take a long view of the inflation rate. This can be readily seen from the difference in the autocorrelation patterns of the excess returns and the inflation series. For high-frequency data
Table I. Annual excess return regressionsa (sample period 1954-91)
aFor the data sources and definitions of the variables see the Data Appendix. ITz is the adjusted R2, DW is the Durbin-Watson statistic, &, &F, ,&, and X L are the chi- squared statistics (with degrees of freedoms in brackets) for testing against residual serial correlation, functional form misspecification, non-normality and heteroscedasticity, respectively. The figures in parentheses are the OLS standard errors, and those in the square brackets give the rejection probabilities. The computations were carried out on the Microfit package, details of which can be found in Pesaran and Pesaran (1991).
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(monthly) the inflation series (computed on the basis of the Producer Price Index) display a much higher degree of serial correlation than do the excess returns. This suggested the use of an annual rate of inflation computed on the basis of monthly averages of the Producer Price Index. In contrast to the inflation effect, we found excess returns responding much more strongly to the term premium variable.’ In line with the results reported in Pesaran (1991), we also found statistically significant negative effects for the lagged change in the 3-month T-bill rate, and the lagged dividend yield. However, we found that once an intercept term and the dividend yield variable were included among the regressors, there were no statistical grounds for the inclusion of a price-earnings variable (fsexp from Citibase) in the regression.
The annual regression reported for the SP 500 portfolio in Table I has a very simple form. In addition to an intercept term, it contains the two-period lagged inflation rate, PIt-2, the lagged dividend yield, YSPt-I, the lagged change in the 3-month interest rate, DBt- l , and the lagged term premium, TERMt- The signs of the coefficients of the inflation rate, the dividend yield and the change in the 3-month interest rate are all consistent with the earlier findings (see, for example, Rozeff, 1984; Fama and Schwert, 1977; Pesaran 1991). When the 1Zmonth rate of change in industrial production was added to this list of regressors its contribution to the predictive power of the excess return equation was marginal, and it did not obtain a statistically significant coefficient.
There is little that can be said a priori about the sign of the coefficient of D13t-~. If it is assumed that the change in nominal interest rates acts as a proxy for the variability in the anticipated real rate of return, then the sign should have been positive for risk-averse investors, rather than negative. However, to the extent that nominal interest rates tend to be high (low) and returns tend to be low (high) at the peak (trough) of the business cycle, the negative sign of the interest rate variable is consistent with the business cycle effect on returns, since it traces the state of the economy and thus the business cycle variations in returns. It is also difficult to explain the sign of the term premium. In the bivariate excess returns regressions for 1953-87 reported by Fama and French (1989) the coefficient of a similar variable changes from a positive sign (for horizons up to 1 year) to a negative sign (for longer horizons), although not all the coefficients in that study are statistically significant. The regressors in the annual regressions reported in Table I are dated earlier as compared with the ones in the Pesaran study, but the regression still explains a sizable proportion of the variations in annual excess return (R2 = 0.59). Note that the estimation period includes the stock market crash of October 1987, although it is true that the effect of the crash on the annual rate of return measured in January 1988 relative to January 1987 is not all that large. Also, none of the diagnostic statistics given in Table I are statistically significant and do not suggest gross misspecifications of the annual regression equation.
Table I also presents the results from a similar regression of excess returns on the Dow Jones industrial portfolio, ERDJt. This regression uses the same regressors as the excess return equation for the SP 500 index, except for the dividend yield variable which is now computed on the basis of the 30 shares contained in the Dow Jones index. The results are very similar to those for the SP 500 portfolio. All the coefficients are highly significant and have the same
Campbell (1987), in his regressions of monthly excess returns, found the coefficient of a similar short-term measure of the term premium (a 2-month rate minus a 1-month rate) to be statistically significant, whereas the coefficient of a longer-term measure (a 6-month rate minus a I-month rate) turned out to be statistically insignificant. On the other hand, Fama and French (1989). in regressions of monthly and quarterly post-war excess returns, obtained significant regression coefficients for a long-term measure of the term premium. The predictive power of the term premium appears to decrease with the time horizon over which excess returns are measured. 6The correlation between the excess return on the SP 500 portfolio and the excess returns on the Dow Jones portfolio, measured at annual frequencies over the period 1954-91, is 0.966.
M. ffashem Pesaran and Allan Timmermann Forecasting Stock Returns 341
signs as in the SP 500 excess return regressions. The overall fit of the regression is, however, somewhat reduced and partly reflects the greater variability of the Dow Jones excess returns as compared to the SP 500 index. In principle, one could search over the space of possible regressors and lag lengths to improve on the Dow Jones regression results in Table I. This is not, however, our purpose in reporting these additional annual results. We are well aware that one should be cautious in interpreting the results in Table I. First, we only use 36 observations. Second, as noted above, there seems to be only weak theoretical rationale for inclusion of some of the regressors in the excess return equation. It is therefore important to see whether our results can be replicated using quarterly or monthly observations.
Quarterly excess returns regressions The regression equation for the quarterly excess returns on the SPSOO portfolio, ERSPf, estimated over the period 1954(1)-1990(4), is presented in Table 11. ’ In arriving at this result we experimented with alternative measures of inflation and interest rates and tried different lag lengths. As in the case of the annual regressions, we found strong statistical evidence in favour of a long measure for the effect of inflation on excess returns. We experimented with a short (monthly) and a long (annual) inflation measure based on the latest published figures for the Producer Price Index and found that the most recently available annual inflation rate, PZ12f-~, performed best. Unlike inflation, changes in interest rates tend to affect excess returns quite
’The quarterly returns on the SP 500 index is computed using the 12-month moving average dividends per share figures published quarterly by the Standard and Poor’s Statistical Service. This procedure smooths out the quarterly dividend figures and may result in a minor measurement error in the computation of quarterly returns. To investigate the importance of this errors in variables problem we carried out the following calculations. For the value weighted CRSP portfolio we computed a similar moving average of dividends based on the quarterly dividends (DIVVW):
DVI = (DIVVW, + DIVVWt- 1 + DIVVWl-2 + DIVVW,-,)/4
and compared the excess return on the value weighted CRSP portfolio computed using DV, with the corresponding measure based on DWVW,. We found that the two measures had a correlation coefficient equal to unity. The reason is that almost all variations in excess returns are due to variations in capital gains.
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quickly (this is in line with the results obtained for the annual regressions). We found a highly significant negative coefficient for 12-month changes in the 1-month T-bill rate, DZ112,- 1. This is the most up-to-date information on the annual change in the 1-month interest rate which is available to the investors at the time of trading.
Our particular choice of the interest rate variable is based on econometric considerations. Since there is no unit root in the process generating excess returns, it is important that the regressors used for predicting the excess returns are themselves either stationary or else form a co-integrating vector. Application of the augmented Dickey-Fuller test shows that we cannot reject the hypothesis of a unit root in the (nominal) interest rates. For the 1-month interest rate (Zl) the twelfth-order augmented Dickey-Fuller statistic computed over the period 1954(1)-1990(4) was -2.19 (-2.87) without a time trend, and - 2.49 (- 3.42) including a time trend. The relevant 95% critical values are given in brackets.'
In contrast to the results at the annual horizon the change in the 12-month industrial production index had significant predictive power for the quarterly excess returns. The negative coefficient of the change in industrial production in the quarterly excess return regression reported in Table I1 is consistent with the general equilibrium result that expected stock returns are low during economic upturns and high during recessions (Balvers et al. 1990).
The diagnostic statistics reported in Table I1 provide significant evidence of departure from the assumption of normally distributed errors, but pass the other tests, namely, of residual serial correlation, functional form misspecification and heteroscedasticity. Since it is a well- established empirical phenomenon that the distribution of stock returns is highly leptokurtic for short return horizons, it comes as no surprise that the residuals from the quarterly excess return regressions are non-normal. Finally note from the regression results that when we go from the annual to the quarterly SP 500 excess return regression the 2' decreases from 0.59 to 0.19. This is in line with the results in the previous studies. See for example, Fama and French (1989).
In order to check the robustness of the quarterly results to the choice of the SP 500 portfolio we ran similar regressions for the excess returns on the Dow Jones industrial portfolio (ERDJ,). We did not experiment to see whether we could obtain a better set of regressors for this portfolio. The purpose of the exercise was solely to see whether the results obtained for the SP 500 portfolio carried over to the Dow Jones portfolio. All the regression coefficients in the excess return equation for the Dow Jones index are statistically significant and have the same signs and are of the same order of magnitudes as those reported for the SP 500 portfolio.' Compared to the quarterly SP 500 regression the 2' decreases from 0.19 to 0.17, but the standard errors of the two regressions are very similar.
Our results also hold for broader classes of portfolios. Using the same set of regressors as in Table I1 we were able to explain 0.185% and 0.163% (after adjustments for the loss in degrees of freedom) of the quarterly variations in the value and equal weighted CRSP indices, respectively. lo Since the CRSP portfolios contain small stocks as well as stocks which are not as liquid as the ones included in the SP 500 or the Dow Jones indices, these results provide further evidence on the robustness of the excess return equations to the choice of the market portfolio.
'See, for example, Dickey and Fuller (1979, 1981). 9The estimated coefficients of the quarterly SP S O 0 excess return equation in Table 1 are within one standard error of the coefficients of the quarterly excess return equation for the Dow Jones index. "To save space these results are not reported here but are available on request from the authors. Using the 1-month rate of inflation instead of the average annual inflation rate in the monthly excess return equation did not produce a statistically significant regression coefficient.
M. Hashem Pesaran and Aiian Timmermann Forecasting Stock Returns 343
Monthly excess return regressions We found that the same basic set of regressors which were capable of predicting excess returns at the quarterly horizon could predict monthly stock returns as well. The lagged values of the dividend yield, the annual measure of inflation, the change in the 1-month T-bill rate, and the 12-month change in the industrial production index all obtained statistically significant coefficients in the monthly excess returns regression. Whereas a 12-month change in the 1-month T-bill rate was included in the quarterly excess return equation, the shorter 1-month change in the 1-month T-bill rate had stronger predictive power for the monthly observations. The included 'long' measure of the inflation rate performed much better than a shorter inflation rate based on more recent trends in producer prices.
The signs of the coefficients in the monthly excess return equations are all consistent with those obtained for the quarterly and annual observations: the yield variable has a positive effect on excess returns, whereas the effect on excess returns of the inflation rate, the change in the 1-month T-bill rate and the rate of change in industrial production are all negative. Term and default premia, the rate of growth in the monetary base, and long measures of changes in interest rates did not obtain significant regression coefficients when included in the equations reported in Table 111. Furthermore, a January dummy did not obtain a statistically significant coefficient in the monthly excess return equations for the SP 500 and the Dow Jones port folios.
The hypothesis that the residuals from the excess-return equation are normally distributed is now rejected even more strongly than in the case of the quarterly regressions. Also, the Rz of the monthly excess return equation for the SP 500 portfolio (0.08) is significantly lower than the similar values for the quarterly (0.19) and the annual horizons (0.59).
" In contrast, the coefficient of the January dummy was highly statistically significant in the excess return regression of the equal-weighted CRSP portfolio. This portfolio puts more weight on small stocks than the Dow Jones or the SP 500 portfolios. Thus our results confirm the link reported in the finance literature between high stock returns in January and the performance of small stocks.
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Business cycle indicators as predictors of excess returns Starting with the regressions reported in the previous section, we performed further experiments with a number of business cycle indicators from the Citibase Data Bank. These indicators include net business formation and composite measures of leading, lagging, and coinciding business cycle variables. We added the lagged rate of change of the 12-month averages of these variables to the regressions reported in Tables 1-111.
In the annual excess-return regressions none of the lagged rates of change in the business cycle indicators proved to be statistically significant when the regression was estimated over the period 1954-91, and this was true for both the SP 500 and the Dow Jones portfolios. However, experimenting with more recent sample periods we were able to find some support for including the leading business cycle indicator in the annual regressions.
In the case of the quarterly regressions we found significant effects for the lagged annual rate of change in the indicator of business formation, DBUS-2, and the lagged annual rate of change in the leading indicator, DLEADt-z. When these variables were added to the basic set of regressors in Table I1 the rate of change in industrial production no longer had a statistically significant coefficient and the default premium, DEFr- 1, proved to be a better measure of the effect of interest rate on excess returns than the DZ112f-1 variable. Excluding the rate of change in industrial production and the 12-month change in the I-month T-bill rate and adding the default premium as well as the leading and the business formation indicators to the list of regressors of the quarterly SP 500 excess return equation, we obtained the estimates -0.82 (-4.52), and 0.47 (2.34) for the coefficients of DLEADt-2 and DBUSI-z respectively. The bracketed figures are the 1-ratios. Adding these two variables to the list of regressors in Table I1 increased the R 2 of the quarterly regression from 0.19 to 0.24. At the quarterly frequency it thus seems that a leading and a business formation indicator are better than the rate of change in industrial production at tracing predictable patterns in stock returns. The rates of change in the lagged and the coinciding business cycle indicators did not have significant effects on quarterly excess returns over the period 1954-90. Very similar results were obtained when we included the business cycle indicators in the quarterly regression for the excess returns on the Dow Jones portfolio.
At the monthly frequency the lagged values of the rate of change in the leading and the business formation indicators both obtained statistically significant coefficients when they were added to the list of regressors in Table 111. In the case of SP 500 portfolio we obtained the estimates of -0.183 (- 3.00), and 0.163 (2.35) for the coefficients of DLEADt-2 and DBUSt-2, respectively. Note that the signs of these regression coefficients are consistent with the signs of similar variables in the quarterly excess return equations. The R 2 of the monthly excess return equations increased from 0.082 to 0.098 and from 0.075 to 0.090 when these business cycle indicators were included in the excess return regressions for the SP 500 and the Dow Jones indices, respectively. Lagging and coinciding business cycle indicators added only marginally to the predictive power of the monthly excess-returns equations.
Non-linear effects of past returns in the excess-returns regressions The literature on the estimation of risk premia on stocks covers a wide variety of approaches. Some attempts have been made to estimate historical measures of the riskiness of the market at a particular point in time based on some average of past volatility in returns. French et al. (1987) estimate ARIMA (1,3) and GARCH models for monthly data for the period 1928-84. The results from these exercises are generally disappointing since the predicted variability from the ARIMA model as well as the GARCH-in-mean estimates of the mean effect of the conditional expectations of variability on monthly returns are not significant at the 5% level.
M. Hashem Pesaran and Allan Timmermann Forecasting Stock Returns 345
In our regression framework we proceeded with a simple exercise. Recognizing that the effect of past shocks on returns may depend non-linearly on the size of the shock as well as on the sign of the shock (negative returns possibly mattering more than positive returns), we experimented with non-linear terms such as the lagged values of the square of returns, p : , and lagged values of p ; = Min(O,pt) and p: = Max(O,pt). We were able to find significant non- linear effects only in the case of the quarterly and monthly regressions. When added to the regressors of the quarterly regressions reported in Table I1 the one- and two-period lagged values of p : obtained significant coefficients in the quarterly excess-return equations. In the quarterly regression for the SP 500 portfolio, p!- I had a positive coefficient of 1.30 (2.30) and ptL2 had a negative coefficient of -0.37 (-2.69) (t-values given in brackets).” The signs of the coefficients are consistent with the notion that a negative return and higher volatility increases the risk premium on stocks. The R2 of the quarterly excess-return equations which include these non-linear effects increased from 0.24 to 0.29 for the SP 500 portfolio and from 0.22 to 0.28 for the Dow Jones portfolio.
In the case of the monthly excess return regressions for the SP 500 portfolio, the variables pt-l , p : - 2 , and pC1 capturing the non-linear effects and the two business cycle variables, DLEADt-z and DBUSt-z, all obtained statistically significant coefficients when added to the regressors reported in Table 111. In this monthly regression the coefficient of p:-1 was -1.32 (-2.38), while the coefficients of p:-z and were 1.31 (2.02), and -0.19 (-2.16), respectively. The R2 of the SP 500 equation which includes the non-linear transformations of returns increased from 0.098 to 0.110. Similar results were obtained when these non-linear effects were introduced in the monthly regression for the Dow Jones portfolio.
2
A NON-PARAMETRIC TEST OF THE PREDICTABILITY OF EXCESS RETURNS
Here we apply the non-parametric predictive failure test recently developed by Pesaran and Timrnerrnann (1992) to the recursive predictions computed on the basis of the excess-return regressions. This test is based on the directional accuracy of the forecasts and compares the proportion of times that the sign of the excess return variable is correctly predicted (irrespective of whether it is positive or negative) to its estimate obtained under the null hypothesis that the predictions and the realizations of excess returns are independently distributed. This predictive failure sign test is distribution free and should be viewed as complementing the regression tests reported in the previous section. l 3 In fact, in a recent study Leitch and Tanner (1991) find that the ranking of forecasts based on directional accuracy (such as the proportion of times that the direction is correctly predicted) matches more closely the ranking based on average profits than do the traditional criteria such as the mean squared forecast errors.
The predictive failure test is particularly suited to test market timing. Standard efficient market models do not rule out the possibility that excess returns on stocks can be predictable. However, they often cannot predict negative excess returns. Consider the classic model studied by Merton (1980):
E ( ~ r + l ( Q t ) = ~ ~ v A R ( p t + l 1 % )
In these quarterly regressions we used DEF,-I (which had a statistically significant coefficient) instead of D1112,- I
(which did not have a statistically significant coeficient) as the interest rate measure. l 3 It can be shown that in the simple 2 x 2 case the non-parametric predictive failure test is asymptotically equivalent to the Henriksson and Merton (1981) test of market timing ability. (See Pesaran and Timmermann, 1993).
346 Journal of Forecasting Vol. 13, Iss. No. 4
where u is a measure of risk aversion and E(. I 62,) and Var(. 162,) are the expectations and variance operators conditional on available information at time t . This model is typically estimated by using a GARCH-in-mean procedure. Since the conditional variance of excess returns is positive and risk-averse investors require a premium to hold the risky asset, this model always predicts that excess returns should be positive. Thus predictions obtained from GARCH-in-mean models will produce a zero value of the predictive failure test statistic. Statistically significant, positive values of the non-parametric predictive failure test can be obtained only when the prediction rule is capable of forecasting accurately periods with negative excess returns. Only in this case will it be possible to outperform the payoffs from a buy-and-hold strategy in the market portfolio.
On the basis of the reported regressions we computed recursive predictions of the excess returns on the SP 500 and the Dow Jones portfolios over the period 1960-90. These recursive predictions utilize only information which is readily available at the time forecasts are made. For example, the prediction of the excess return for January 1960 is based on the monthly excess-return regression using monthly observations over the period 1954( 1)- 1959( 12). The prediction for February 1960 is then carried out by re-estimating the regression now over the period 1954(1)-1960(1), and so on. The actual values of the annual excess returns and the corresponding recursive predictions for the SP 500 portfolio are displayed in Figure 1. Except in 1976 the recursive predictions follow the movements of the actual excess returns remarkably well. It is particularly of interest that the recursive procedure is capable of accurately predicting the large negative values of the excess returns in 1969, 1974-5 and 1982. Also, the recursive predictions at the quarterly and monthly frequencies are often capable of predicting large negative excess returns (Figures 2 and 3).
Figure 2. Actual and recursive predictions of quarterly excess returns (SP 500)
. 1 m
.En994
- .832719
- .u855 1968nl 1%7n10
ERSP
Figure 3. Actual and recursive predictions of monthly excess returns (SP 500)
p m p . . . . . . . . . . . . . . . . . . . . .
348 Journal of Forecasting Vol. 13, Zss. No. 4
The values of the predictive failure test statistic computed on the basis of the recursive predictions are displayed in Table IV. Under the null hypothesis of no market-timing skills the predictive failure test statistic is asymptotically distributed as a standardized normal variate. Since we are interested in situations where the proportion of correctly predicted signs is larger than the value of this proportion under the null hypothesis, only positive and statistically significant values of the predictive failure test provide evidence of market-timing skills. Hence the appropriate procedure is to apply a one-sided test. The results in Table IV clearly show that a statistically significant proportion of the signs of the realized excess returns are correctly predicted by all excess return regressions over the period 1960-90. The evidence is especially strong at the annual frequency, where 81% of the signs are correctly predicted. Due to the smaller number of observations in the sub-samples the values of the predictive failure tests over the periods 1960-69, 1970-79 and 1980-89 are not as high as the equivalent values for the entire sample 1960-90. Nevertheless, the excess-return equations always predict at least 55% of the signs correctly.
Table IV. The sign test statistics for the recursive predictions based on the excess return regressions
Proportion of correctly Predictive Number
predicted failure test of Frequency signs (Yo) statistic switches
Annual SP 500 (1960-90) DJ (1960-90)
SP 500 (1960-69) SP 500 (1970-79) SP 500 (1980-89) SP 500 (1960-90) DJ (1960-69) DJ (1970-79) DJ (1980-89) DJ (1960-90)
SP 500 (1960-69) SP 500 (1970-79) SP 500 (1980-89) SP 500 (1960-90) DJ (1960-69) DJ (1970-79) DJ (1980-89) DJ (1960-90)
Quarterly
Monthly
80.6 71.0
65.0 62.5 60.0 62.1 65.0 57.5 62.5 62.1
56.7 58.3 60.0 58.1 55.8 56.7 60.8 57.3
3 .Ma 2.54a
1 S O 1 S O 0.50 2.09b 1.62 1.05 0.59 2.22b
1.39 1 .!Job 1.42 2.76a 1.09 1.47 2.21 2.61 a
11 13
5 9
1 1 26
7 7 9
24
16 25 32 77 17 29 34 85
The results are based on the recursive predictions from the SP 500 and Dow Jones (DJ) excess return regression models given in Table 1. 'Significance at the 1 % level. bSignificance at the 5 % level.
M. Hashem Pesaran and Allan Timmermann Forecasting Stock Returns 349
THE TRADING RESULTS
The trading rule In the finance literature the efficient market hypothesis is often formulated in terms of the impossibility of constructing a trading rule, based on publicly available information, which is capable of yielding positive excess profits (discounted at an appropriate risk adjusted rate). Jensen (1978) puts it this way: ‘A market is efficient with respect to information set at if it is impossible to make economic profits by trading on the basis of information set at.’ Specifically, if a, denotes the publicly available information at time t , then if at cannot be exploited profitably in a trading rule the market is said to be efficient in a semi-strong form (see also the recent survey by LeRoy, 1989).
The classical trading strategy utilizes a filter rule (Alexander, 1961; Fama and Blume, 1966). This rule is based on the belief that, due to the gradual leakage of information, detectable short-run trends appear in the level of stock prices and that these trends can be exploited profitably. The rule, in its most common version, tells the investor to go long in a portfolio when the price of the portfolio increases by at least x% from a preceding low ( x being the size of the filter), and to go short if the price drops by at least x% from its previous peak. Price changes that are numerically less than x% from preceding lows or highs do not trigger off any trades. l4
This strategy is, however, subject to a number of criticisms: First, it only exploits a small part of the publicly available information. As we argued above, it is by now generally accepted that one can predict a substantial proportion of the variations in excess returns, and there is no reason to restrict the set of regressors to past prices. Second, the filter rule with the highest expected return often results in very frequent trading (corresponding to a small x ) , and this in turn implies that the excess returns, net of transaction costs and costs of supervision, are either lower than the ones from a simple buy-and-hold strategy or only insignificantly better (cf. Fama and Blume, 1966, pp. 238-9). If one alternatively uses a large filter, the funds may be out of the market for a large proportion of the time.”
The other strand in the literature exploits more elaborate regression methods. Breen et al. (1990) run a regression of a qualitative measure of excess returns defined as yt = Max(0, pr ) /pr , on the lagged value of the yield on the AA rated long-term corporate bond, the book value of the market index (taken to be SP 400) relative to its market value, and the earnings price ratio of the market index. This approach may be inefficient since the regression throws away information due to the qualitative transformation of the excess return variable, pr. The trading rule of Breen et al. (1990) is to stay in the stock index portfolio if the predicted value of yt > 0.5, otherwise to stay in treasury bills. Breen et al. (1990), however, do not allow for transaction costs which may be substantial in the case of the switching portfolio that they consider.
Here we employ the regression results reported above to compute recursive predictions
l4 This strategy is the opposite of one based on mean reversion in stock prices around an underlying trend. Such a strategy would tell the investor to go short if the market was, say, x% above the underlying trend, and to go long if the market was x% below the underlying trend. The differences in the strategies are due to the different models assumed for stock price movements. ”Bulkley and Tonks (1989), using a filter rule of the mean reversion type, derive their optimal filter, x, recursively. They choose the filter size, x, at time t, which, if applied to the data up to time t , would have resulted in the highest return. Unfortunately, the use of such an updating rule can result in filter sizes that are dominated by large price changes in the past. See Table I1 in the 1987 version of their paper, where they show that x falls from 0.18 to 0.12 in 1937 and stays at that value from then on.
350 Journal of Forecasting Vol. 13, Iss. No. 4
which we use to construct a switching portfolio similar to that of Breen et al. (1990). A particularly simple rule is adopted: hold the market portfolio if the recursively predicted excess return is positive, otherwise hold government bonds with a maturity equal to the duration of the trading horizon. We do not claim optimality for such a rule. To derive an optimal trading strategy requires knowledge of the investor’s utility function and the probability distribution of excess returns. We avoid the need for the specification of such distributional assumptions and focus our analysis on the simple switching strategy based only on the predicted sign of the excess return variable. We do, however, explicitly allow for transaction costs. The switching strategy has the further advantage that it avoids the problem associated with bankruptcy and, since no gearing is used, the payoffs of the switching portfolio are directly comparable to those of the market portfolio.
In constructing a trading rule we are careful in only using information which is readily available to potential investors at the time when the investment decision is taken. Since the variability in stock prices within a month or even within a single day can be very large, it is important to consider trading rules that specify the precise time at which the trade is to take place, i.e. a particular time within a particular day. It is, for example, not appropriate to use average monthly prices to evaluate or construct trading rules.
Transaction costs l 6
Transaction costs of dealing in stocks consist of two parts, namely the costs associated with the bid-ask spread and the commission (see, for example, Demsetz, 1968). The total cost of transaction has varied over the period 1960-90 and it varies with the size of the transaction and with the specific share that is being transacted. Since we are using closing prices in our study, which could equally well be bid or ask prices, and since the commission is independent of whether stocks are being bought or sold, we assume that the transaction costs for stocks are symmetric in the sense that the size of the costs do not depend on whether the investor is buying or selling. The size of the transaction costs” is assumed to be proportional to the sum traded. The marginal cost of transactions is assumed to be c1 (fixed) when the investor trades in stocks and c2 (fixed) when the investor buys bonds. l 8 No transaction costs are incurred when bonds or treasury bills are sold (cashed) on their maturity. The costs are paid at the time of the transaction. Let
Wf = funds available to the investor at time t , Nr = number of shares held at time t (after trading), Pf = price of shares at time t , Dt = dividends per share paid during the period, rt = the rate of return on bonds in period t , Bf = position in bonds at time t (after trading),
I60n the importance of transaction costs for the frequency and volume of trade in an equilibrium asset pricing model, see Constantinides (1986) and Dumas and Lucian0 (1991).
In the following the term transaction costs is referred to as comprising the cost associated with the bid-ask spread as well as the commission. ‘*In practice, it is likely that the marginal transaction cost may vary over time, and this could be of some importance in the case of extreme volatility in the market as happened during the October 1987 crash. In view of the small number of transactions associated with the switching portfolio, we do not expect such variability in transaction costs to be particularly important in our application.
M. Hashem Pesaran and AIIan Timmermann Forecasting Stock Returns 35 1
Zt+l = indicator variable taking the value 1 or 0 according to the predicted sign of excess returns in period t + 1:
& + I = 1 , if ,,%+I > 0, = 0, otherwise,
where & + I denotes the recursive prediction of the excess returns on the market portfolio in period t + 1 computed with respect to the information set at time t .
At time t the investor allocates the funds into either bonds or stocks according to the prediction of the sign of the excess returns for period t + 1 , i.e. according to the value of It+ 1
defined above. The investor faces a budget constraint and pays transaction costs according to the formulae:
In period t + 1 the investor allocates the available funds according to the prediction of excess returns for period t + 2 based on the publicly available information on the last trading day of period t + 1 ( Z t + 2 ) . By reallocating the funds the investor incurs some transaction costs, the size of which is determined by the composition of the investor’s existing portfolio in bonds (Bt) or stocks (Nt) and by the selected portfolio composition for period t + 1 . Four cases need to be distinguished:
Case 1
Z t + l = 1 and Z t + 2 = 1 (reinvest cash dividends in stocks) Nt+ 1 = N, + NtDr(1 - c1 I/&+ 1
Bt+l=O
Case 3 & + I = 0 and Z1+2 = 1 (bonds mature, buy stocks)
N+I = (1 - cl)Bt(l + rt)/Pt+l Bt+1=0
Case 4 & + I = 0 and
Bt+l = ( 1 - cz)(l + rt)Bt
Zt+2 = 0 (bonds mature, buy bonds) Nt+l=O
352 Journal of Forecasting Vol. 13, Iss. No. 4
The funds available at time t + 2 are given by
W r + 2 = N t + l ( P t + ~ + D 1 + 1 ) + B t + l ( l + r r + i ) (6)
These rules can be readily extended to the subsequent time periods. l 9
To get an idea of the likely values of the cost parameters c1 and cz we consulted a number of dealers in the security market concerning the bid-ask spread and checked with advertised commissions. In general, the trading fee for bonds is either zero or very low ($25 for a trade of any size), For this reason, we considered the values of 0.0% and 0.1% for c2. The bid-ask spread for frequently traded stocks is typically 1/8 of a point. Since we are using closing prices, the average cost associated with the bid-ask spread for stocks quoted at prices from 40 points to 120 points will lie in the range from 2/10 of a per cent and 5/100 of a per cent of the transacted funds. Brokerage fees for stocks is composed of a cent per share and a percentage of the principal value. Generally, commissions on stocks range from 0.05 of a per cent to 2% for trades of a reasonable size ($3000 and upwards) in the same share. A typical quotation involves a commission of 0.2% for buying 300 shares priced at $50 each. To limit the transaction costs to this percentage, a trade in the 500 individual stocks included in the SP 500 basket would thus require a capital of $7.5 million. This is not out of the reach of institutional and even some private investors. We report on trading results for c1 equal to 0070, 0.5070, and 1%. The zero transaction cost scenario should be viewed as a benchmark, and allows our trading results to be compared with those given by Breen et al. (1990). The case with c1, equal to 0.5% can be regarded as a low-cost scenario, whereas the scenario with CI equal to 1% characterizes a high cost scenario. The latter is relevant to a small private investor.20
Payoffs from trading based on the recursive regressions A number of summary statistics relevant to the trading results are given in Tables V-X. It is clear that the annual and quarterly switching portfolios pay a higher mean return than the bonds and the market portfolios. This conclusion holds even when the high transaction cost scenario is considered. However, mean returns on the monthly switching portfolios do not exceed mean returns on the market indexes when transaction costs are high. Assuming zero transaction costs the differences in annualized mean returns between the switching portfolios and the market portfolios based on the SP 500 index were 1.9070 (annual), 2.2% (quarterly), and 2.3% (monthly). Under the high transaction costs scenario these differences amounted to 1.5% (annual), 1.1 To (quarterly), and -1 .OVo (monthly). 2 1
Another interesting aspect of the trading results is in the comparison of the mean returns of the switching portfolios across the three trading frequencies. Tables V, VII, and IX show that the mean returns on the quarterly and monthly switching portfolios based on the SP 500 index exceed the mean return on the monthly switching strategy when transaction costs are zero. In the low transaction costs scenario the annual and quarterly switching portfolios based on the SP 500 portfolio pay a very similar mean return, while the mean return on the monthly switching portfolio is somewhat lower. When transaction costs are high the annual switching strategy pays the highest mean return. These results indicate that it only pays to trade more frequently than at the annual interval if transaction costs can be kept at a low level (less than
I9In particular, note that dividends paid during the holding period are not continuously reinvested, but are accumulated until the end of the holding period at which time they are invested in T-bills/bonds or stocks. 20For institutional size trades Beckers (1992) estimated total trading costs in the USA of 30-45 basis points. ''In the case of the market and T-bill portfolios, the transaction costs matter only for the reinvestment of the dividends on the market portfolio and the roll-over of the T-bills at maturity. In Tables V-X this also explains why there are three columns under the market portfolio and two columns under the T-bills.
% Ta
ble
V.
Perf
orm
ance
mea
sure
s of
the
SP
500
switc
hing
por
tfol
io r
elat
ive
to th
e m
arke
t po
rtfo
lio a
nd T
-bill
saSb
(ann
ual
resu
lts:
1960
-90)
Mar
ket
Port
folio
s Switc
hing
Tra
nsac
tion
cost
(To)
St
ocks
T-
bills
A
rithm
etic
mea
n re
turn
(Yo)
SD o
f re
turn
(To
) Sh
arpe
's in
dex
Tre
ynor
's in
dex
Jens
en's
inde
x
Wea
lth a
t en
d of
per
iod'
0.0
10.7
8 13
.09
0.31
0.
040
-
-
1913
0.5
10.7
2 13
.09
0.30
0.
040
-
-
1884
1 .o
10.6
7 13
.09
0.30
0.
039
-
-
1855
0.0
0.0
12.7
0 7.
24
0.82
0.
089
0.04
5 (4
.63)
38
33
0.5
0. I
12.4
3 7.
20
0.79
0.
085
0.04
3 (4
.42)
35
59
%
9 T-
Bill
s %
-
2 0.
1 0.
0 0.
1 a" (D
1 .o
-
12.2
1 6.
75
6.64
3
7.16
2.
82
2.82
9
3
Q
0.76
b
0.08
1 -
-
0.04
1 -
-
(4.2
5)
-
-
-
-
$ 33
46
749
726
"The
sw
itchi
ng p
ortf
olio
is b
ased
on
recu
rsiv
e re
gres
sion
s of
exc
ess r
etur
ns on
the
chan
ge in
the
3-m
onth
inte
rest
rat
e, t
he te
rm p
rem
ium
, th
e in
flatio
n ra
te,
and
the
divi
dend
yie
ld (
see
Tabl
e I).
The
sw
itchi
ng ru
le a
ssum
es t
hat
port
folio
sel
ectio
n ta
kes
plac
e on
ce a
yea
r at
clo
sing
tim
e on
the
last
trad
ing
day
of J
anua
ry.
bFor
a de
scrip
tion
and
the
ratio
nale
beh
ind
the
vario
us p
erfo
rman
ce m
easu
res
used
in
this
tab
le,
see,
for
exa
mpl
e, L
evy
and
Sarn
at (
1984
, Ch.
15)
. T
he m
arke
t po
rtfo
lio d
enot
es a
buy
-and
-hol
d st
rate
gy i
n th
e SP
500
inde
x. T
-bill
s de
note
a r
oll-o
ver
stra
tegy
in
12-m
onth
T-b
ills.
'S
tart
ing
from
$10
0 in
Jan
uary
196
0.
Tabl
e V
I. Pe
rfor
man
ce m
easu
res o
f th
e D
ow J
ones
sw
itchi
ng p
ortf
olio
rel
ativ
e to
the
mar
ket p
ortf
olio
and
T-b
illsa
Sb (a
nnua
l res
ults
: 19
60-9
0)
Port
folio
s M
arke
t Sw
itchi
ng
T-B
ills
Tra
nsac
tion
cost
(Yo)
St
ocks
0.
0 T-
bills
-
Arit
hmet
ic m
ean
retu
rn (
To)
10.1
1 SD
of
retu
rn (
To)
15.1
1 Sh
arpe
's in
dex
0.22
Tr
eyno
r's i
ndex
0.
034
Jens
en's
inde
x -
Wea
lth a
t en
d of
per
iod'
-
1486
0.5
10.0
5 15
.12
0.22
0.
033
-
-
-
1463
1 .o
10.0
0 15
.13
0.22
0.
033
-
-
-
1440
0.0
0.0
11.4
4 8.
82
0.53
0.
077
0.03
6 (2
.89)
26
33
0.5
0.1
11.1
3 8.7
3 0.
50
0.07
2 0.
033
(2.6
8)
2415
1 .o
0.1
10.8
7 8.
67
0.48
0.
067
0.03
0 (2
.51)
22
5 1
-
0.0
6.75
2.
82
-
0.1
6.64
2.82
a-c
Se
e th
e no
tes
to T
able
V.
w
lA
w
Tabl
e V
II.
Perf
orm
ance
mea
sure
s of
the
SP
500
switc
hing
por
tfol
io r
elat
ive
to th
e m
arke
t por
tfol
io a
nd T
-bill
sagb
(qua
rter
ly re
sults
: 19
60( 1
)-9O
(4))
w
VI
Port
folio
s T-
Bill
s P
M
arke
t Sw
itchi
ng
Tran
sact
ion
cost
(To
) 6
T-b
ills
-
-
-
0.0
0.1
0.1
0.0
0.1
3 5 -
-
-
(2.8
3)
(2.6
0)
(2.3
3)
-
-
8
-
-
E:
Stoc
ks
0.0
0.5
1 .o
0.0
0.5
1 .o
Ari
thm
etic
mea
n re
turn
(To
) 10
.87
10.8
2 10
.77
13.0
4 12
.43
11.9
1 6.
49
6.06
‘
SD o
f re
turn
(To)
15
.78
15.8
1 15
.83
11.0
9 11
.06
11.0
4 2.
84
2.83
%
Sh
arpe
’s i
ndex
0.
28
0.30
0.
30
0.60
0.
58
0.53
-
-
%
Trey
nor’
s in
dex
0.044
0.04
8 0.
047
0.13
3 0.
129
0.12
1 -
-
Jens
en’s
inde
x -
-
-
0.05
1
0.04
8 0.
043
-
-
3
Wea
lth a
t en
d of
per
iod‘
17
73
1745
17
18
3922
32
69
283
1 69
4 61
3 0s
0
aThe
switc
hing
por
tfol
io i
s ba
sed
on r
ecur
sive
reg
ress
ions
of
exce
ss r
etur
ns o
n an
inte
rcep
t te
rm,
the
lagg
ed d
ivid
end
yiel
d, t
he l
agge
d 12
-mon
th in
flatio
n ra
te,
the
lagg
ed c
hang
e in
the
1-m
onth
inte
rest
rat
e, a
nd t
he l
agge
d ch
ange
in i
ndus
tria
l pr
oduc
tion.
The
sw
itchi
ng r
ule
assu
mes
tha
t po
rtfo
lio s
elec
tion
take
s pl
ace
once
a
year
at
clos
ing
time
on th
e la
st t
radi
ng d
ay o
f ea
ch q
uart
er. See
also
Tab
le 11
. bF
or a
desc
riptio
n an
d th
e ra
tiona
le b
ehin
d th
e va
rious
per
form
ance
mea
sure
s us
ed in
this
tabl
e, s
ee, f
or e
xam
ple
Levy
and
Sar
nat (
1984
, Ch.
15)
. T
he m
arke
t por
tfol
io
deno
tes
a bu
y-an
d-ho
ld s
trate
gy i
n th
e SP
500
inde
x. T
-bill
s de
note
a r
oll-o
ver
stra
tegy
in
3-m
onth
T-b
ills.
‘S
tarti
ng f
rom
$10
0 in
Dec
embe
r 19
59.
Tabl
e V
III.
Per
form
ance
mea
sure
s of
the
Dow
Jon
es s
witc
hing
por
tfol
io r
elat
ive
to t
he m
arke
t po
rtfo
lio a
nd T
-bill
saSb
(qua
rter
ly r
esul
ts:
1960
( 1)-9
O(4
))
Mar
ket
Port
folio
s Switc
hing
T-
Bill
s
Tran
sact
ion
cost
(To)
St
ocks
T-
bills
A
rith
met
ic m
ean
retu
rn (
To)
SD o
f re
turn
(To
) Sh
arpe
’s i
ndex
Tr
eyno
r’s
inde
x Je
nsen
’s in
dex
Wea
lth a
t end
of
peri
od‘
0.0
10.2
7 16
.65
0.23
0.
038
-
-
-
1448
0.5
10.2
1 16
.67
0.25
0.
041
-
-
-
1425
1 .o
0.0
-
0.0
10.1
6 12
.35
16.7
0 12
.24
0.25
0.
48
0.04
1 0.
110
-
0.044
-
(2.3
3)
1401
31
01
0.5
0.1
11.7
3 12
.28
0.46
0.
107
0.04
1 (2
.12)
26
02
1 .o
0.1
1 1.2
5 12
.30
0.42
0.
099
0.03
6 (1
.88)
22
76
-
0.0
6.49
2.
84
-
0.1
6.06
2
2.83
2
-
613 -
5
a-‘S
ee t
he n
otes
to
Tab
le V
II.
Tabl
e IX
. Pe
rfor
man
ce m
easu
res o
f th
e SP
500
sw
itchi
ng p
ortf
olio
rel
ativ
e to
the
mar
ket
port
folio
and
T-b
illsa
Sb (m
onth
ly re
sults
: 19
60(1
)-90(
12))
Mar
ket
Port
folio
s Switc
hing
T-
Bill
s
Tra
nsac
tion
cost
(To)
St
ocks
T-
bills
A
rithm
etic
mea
n re
turn
(To
) SD
of
retu
rn (
To)
Shar
pe’s
inde
x Tr
eyno
r’s
inde
x Je
nsen
’s in
dex
Wea
lth a
t en
d of
per
iod‘
0.0
10.9
0 15
.82
0.3 1
0.
049
-
-
-
784
0.5
10.8
5 15
.85
-
0.38
0.
061
-
-
756
1 .o
10.7
9 15
.87
-
0.38
0.060
-
-
728
0.0
0.0
13.0
8 10
.84
0.65
0.
103
0.04
9 (3
.25)
39
22
0.5
0.1
11.1
4 10
.82
0.59
0.
097
0.03
7 (2
.36)
22
89
1 .o
0.1
9.74
10
.80
0.46
0.
080
0.02
5 (1
.51)
15
42
-
-
0.0
0.1
6.03
4.
76
2.66
2.
63
-
-
-
-
-
-
-
-
608
419
“The
switc
hing
por
tfol
io i
s ba
sed
on r
ecur
sive
reg
ress
ions
of
exce
ss re
turn
s on
an
inte
rcep
t ter
m,
the
lagg
ed d
ivid
end
yiel
d, t
he l
agge
d 12
-mon
th in
flatio
n ra
te,
the
lagg
ed c
hang
e in
the
1-m
onth
T-b
ill ra
te, a
nd th
e la
gged
cha
nge
in in
dust
rial
pro
duct
ion.
The
switc
hing
rule
ass
umes
that
por
tfol
io s
elec
tion
take
s pl
ace
once
a m
onth
at
clo
sing
tim
e on
the
last
tra
ding
day
of
each
mon
th.
See
also
Tab
le 1
11.
bFor
a d
escr
iptio
n an
d th
e ra
tiona
le b
ehin
d th
e va
rious
per
form
ance
mea
sure
s us
ed i
n th
is t
able
, se
e, f
or e
xam
ple,
Lev
y an
d Sa
rnat
(19
84,
Ch.
15)
. T
he m
arke
t po
rtfo
lio d
enot
es a
buy
-and
-hol
d st
rate
gy i
n th
e SP
500
inde
x. T
-bill
s de
note
a r
oll-o
ver
stra
tegy
in
1-m
onth
T-b
ills.
‘S
tart
ing
from
$10
0 in
Dec
embe
r 19
59.
Tab
le X
. Pe
rfor
man
ce
mea
sure
s of
th
e D
ow
Jone
s sw
itchi
ng p
ortf
olio
re
lativ
e to
th
e m
arke
t po
rtfo
lio
and
T-b
illsa
*b (m
onth
ly r
esul
ts:
1960
( 1)-
1990
( 12)
)
Mar
ket
Port
folio
s Switc
hing
T-
Bill
s ~~
~~
~ ~~
~~~
~ ~
~ ~
-
~
Tra
nsac
tion
cost
(Yo)
St
ocks
0.
0 0.
5 1 .o
0.
0 0.
5 1 .o
T-
bills
-
-
-
0.0
0.1
0.1
0.0
0.1
Arit
hmet
ic m
ean
retu
rn (
To)
10.3
0 10
.24
10.1
9 13
.41
11.3
2 9.
79
6.03
4.
76
SD o
f re
turn
(To
) 16
.70
16.7
2 16
.74
12.2
7 12
.21
12.1
8 2.
66
2.63
Sh
arpe
’s in
dex
0.26
0.
33
0.32
0.
60
0.54
0.
41
-
-
Trey
nor’
s in
dex
0.04
3 0.
055
0.05
4 0.
139
0.12
4 0.
097
-
-
Jens
en’s
inde
x -
-
-
0.05
8 0.
045
0.03
0 -
-
-
-
-
(3.0
3)
(2.3
1)
(1.5
3)
-
-
Wea
lth a
t en
d of
per
iod‘
14
58
1434
14
10
4150
23
22
1502
60
8 41
9
-
-
See
the
note
s to
Tab
le IX
.
356 Journal of Forecasting VOI. 13, rss. NO. 4
0.5%). In the case of the switching portfolios based on the Dow Jones index the monthly strategy pays the highest mean return when transaction costs are zero, whereas the quarterly switching portfolio pays a higher mean return than the monthly and the annual switching portfolios in the low and the high transaction cost regimes. The standard deviations of the monthly and quarterly switching portfolios exceed the standard deviation of the annual switching portfolio.
Turning now to the different risks associated with holding different portfolios, it is clear from Tables V-X that the switching portfolios not only pay substantially higher returns over the period, they also have a much lower standard deviation than the market portfolios. This result continues to hold even under the high transaction cost scenario. Thus at the quarterly and annual frequencies the switching portfolios dominate the market portfolios by the mean- variance criterion. At the monthly frequency the large number of switches between bonds and stocks reported in Table IV implies that the mean returns on the switching portfolios are smaller than those on the market portfolios in the high transaction cost scenario. 22
We considered the sensitivity of our results with respect to the choice of the regressors in the excess-return equations. Table XI gives the values of the funds at the end of the trading period as well as the mean and standard deviation of the returns on the SP 500 switching portfolios based on different sets of regressors. Assuming zero or low transaction costs these mean returns are all higher than the mean return on the market portfolios. In the case with high transaction costs the third set of regressors used at the monthly interval results in a lower mean return than the market portfolio. Since interest rates tend to move together over time it is not surprising that our results are particularly robust with respect to the particular set of interest rates included in the regressions.
Finally, we computed the payoffs from trading in the SP 500 index over the sub-samples 1960-69, 1970-79, and 1980-89 (see Table XII). The list of regressors were identical to those in the excess return regressions summarized in Tables 1-111. The quarterly and annual switching portfolios outperform the market portfolio during each and every one of the three decades. Table XI1 also shows that, at the monthly frequency, the mean return on the switching portfolio was lower than the mean return on the market index over the period 1960-69. This picture changes during the volatile 1970s and 1980s.
Performance measures: the Sharpe, Treynor, and Jensen indices Here we apply the various measures of portfolio performance proposed in the finance literature, namely the Sharpe (1966) index, the Treynor (1965) index, and the Jensen (1968) index to the switching and the market portfolios. All these indices measure the mean-variance efficiency of the portfolio under consideration. The Sharpe and Treynor indices are defined by (E(R,) - r ) / u , and (E(R , ) - r ) / & respectively, where E(R,) stands for the mean return of portfolio P , r is the risk-free interest rate, a, is the standard deviation of the return on portfolio P, and 0, is the beta of portfolio P. Portfolio P is then said to dominate the market
22 The reported values of the mean returns on the switching portfolios represent lower bounds since it may be possible to improve the trading results by taking transaction costs into account in the design of the trading rule. For instance, it may be possible to improve the results from trading if a trade only takes place provided that the predicted value of excess returns exceeds transaction costs. To investigate this issue we computed pay-offs from a strategy which switches between bonds and stocks only if the predicted value of excess returns happens to exceed the assumed value for the transaction costs. Using this switching scheme, we obtained the average annual mean returns of 10.70 (11.96) and 10.63 (11.74) for the monthly SP 500 portfolio under the low and the high transaction costs scenarios. The standard errors of the mean returns are in brackets. Thus, compared to the results reported in Table IX, this switching scheme yields a higher mean return when transaction costs are high and a lower one when transaction costs are low.
Tab
le X
I. S
ensi
tivity
of
the
payo
ffs
on th
e sw
itchi
ng p
ortf
olio
s ba
sed
on t
he S
P 50
0 po
rtfo
lio w
ith r
espe
ct t
o di
ffer
ent
sets
of
regr
esso
rs
-~
~~
Reg
ress
ors
~~~
~
Tra
nsac
tion
cost
s Z
ero
Low
H
igh
End
wea
lth
End
wea
lth
End
wea
lth
($)
Mea
n SD
(8
M
ean
SD
($1
Mea
n SD
Ann
ual
Inpt
, Y
SP( -
l), PZ( - 2
), TE
RM
( -l)
, D
EF
( -1)
In
pt,
YSP(
-l),
P
I( -2
), D
Z1(-1
), TE
RM
( -1)
In
pt,
YSP
( -l)
, PZ( -
2), D
Z6(-1
), TE
RM
( -1)
Q
uart
er&
In
pt,
YSP
( -l)
, Il
(-l)
, 11
2(-
2),
Z12(
- 3)
In
pt,
YSP
( -l)
, PZ( -
2), D
IP( -
2), D
Z312
(-1)
Inpt
, Y
SP(-
l),
PZ
(--2
), D
ZP(
-2),
Zl(-
1),
Zl(
-6).
Mon
thly
In
pt,
YSP
( -l)
, Z1
(-1),
BA
A( -
1)
Inpt
, Y
SP(-
l),
PI(
-2),
D
ZP(
-2),
DE
F(-
l)
Inpt
, Y
SP
(-l)
, PZ
(-2)
, D
ZP(
-2),
DZ3
1(-1
)
2662
35
10
2929
5005
34
84
4791
4835
36
73
3998
11.4
8 8.
49
2474
11
.22
8.55
12
.39
7.42
32
24
12.0
9 7.
42
11.7
3 7.
20
2737
11
.48
7.15
14.0
8 12
.59
3852
13
.08
12.1
0 12
.61
10.5
5 30
30
12.1
1 10
.51
14.0
0 12
.98
3569
12
.92
12.8
0
13.8
2 10
.74
3343
12
.50
10.9
7 12
.94
12.0
3 29
85
12.2
2 12
.29
13.1
7 11
.13
2350
11
.27
11.2
3
2326
30
02
2600
3212
27
3 1
2799
2153
26
88
1582
11.0
1 8.
61
11.8
3 7.
43
11.2
9 7.
11
12.3
8 11
.60
11.7
3 10
.47
12.0
3 12
.62
11.8
1 11
.04
11.8
6 12
.41
9.89
11
.38
Not
e: N
et f
unds
giv
e th
e va
lue
at th
e en
d of
the
per
iod
(199
0) o
f in
vest
ing
$100
at
the
begi
nnin
g of
the
per
iod
(196
0).
3 2 3 3 3
3
350 Journal of Forecasting Vol. 13, Zss. No. 4
Table XII. Returns on the portfolios across subsamples
Market index Mean SD
Annual 1960-69 8.13 10.93 1970-79 8.29 16.60
Quarterly 1960-69 8.62 14.32 1970-79 7.39 19.18
Monthly 1960-69 8.63 14.36 1970-79 7.42 19.24
1980-89 16.17 11.34
1980-89 18.00 12.64
1980-89 18.05 12.66
Switching portfolio Mean SD
Bonds Mean SD
11.04 6.31 10.20 5.41 17.33 8.33
9.03 9.65 12.09 11.22 19.70 9.80
8.26 9.04 11.45 9.30 20.28 11.58
4.27 1.08 6.35 1.69 9.48 2.64
3.96 1.28 6.28 1.65 9.10 2.75
3.71 1.32 6.02 1.80 8.23 2.66
Note: The mean is computed arithmetically. The calculations are for the case with zero transaction costs. All statistics are based on annual payoffs. The recursive predictions used to compute the payoffs on the switching portfolios were based on observations starting 5 years prior to the beginning of the decades (i.e. in 1955, 1965, and 1975).
portfolio in the mean-variance sense if its Sharpe or Treynor index is higher than that of the market portfolio. The Jensen index is the intercept in the regression of ( R P - r ) on ( R m - r ) , where R m is the return on the market portfolio. Portfolio P is said to be superior t o the market portfolio if this intercept is positive and statistically significant. 23
The three performance measures for the market and the switching portfolios for the annual, quarterly, and monthly trading strategies are summarized in Tables V-X. The measures support our general conclusion that the switching portfolio offers a better mean-variance trade- off than the market portfolio, even under the high transaction cost scenario. In particular, note that the Jensen measure is statistically significant at the 5 % level at all frequencies except for monthly trading under the high transaction cost scenario.
The trade-off between risk and return At this stage it is reasonable to ask whether our trading results, which in almost all cases give a smaller standard deviation for returns based on the switching portfolio than the market portfolio, is an artifact of mixing the market portfolio with the T-bills which have much smaller standard deviations.
Consider the random variables x- ( p X , u : ) and y - (c~y,u$) where x and y may be correlated. Construct a portfolio z = w x + (1 - w)y, where the weight w is a discrete random variable distributed independently of x and y, and takes the value of 1 with probability PO, (0 < po < 1) and the value of 0 with probability (1 --PO). Hence z is a randomly chosen combination of the two underlying random variables x and y. Suppose that y represents the returns on stocks and x the returns on bonds such that py > px and u$ > a:. It is easily seen
23See Levy and Sarnat (1984, Ch. 15) for further details cowering these indices.
M. Hashem Pesaran and Allan Timmermann Forecasting Stock Returns 359
that 24
E ( z ) = Pz = POCLX + (1 -Po)&
( 8 ) Var(z) = at = p o d + (1 - PO)U; + PO( 1 - P O ) GY - p x )
Using equation (7), we have pz - py =p0(pLx - p Y ) , and since by assumption px < pY, it therefore follows that pz c pY, and the mean return on a switching portfolio constructed in a random fashion cannot exceed the mean return on the market portfolio. The randomly constructed switching portfolio will not be necessarily less risky than the market portfolio either. The condition under which u: > u; is given by (0 < PO C 1):
(7)
and 2
Applying these formulae to annual returns on SP 500 (& = 10.78, L?~ = 13.09) and the T-bills GX = 6.75, CX = 2.82) and choosing PO in accordance with its value in the switching strategy (0.29), we get a portfolio, t, with the mean return 9.61, which is less than the mean return on the market portfolio and the standard deviation 11.28. This is clearly dominated by the payoff from our switching portfolio which has an arithmetic mean of 12.70 and a standard deviation of 7.24. Hence our results are not just due to a random effect from mixing two portfolios. We have actually succeeded in constructing a switching portfolio that displays a more favourable trade-off between the standard deviation and the mean of returns as compared to the market portfolio.
CONCLUSIONS
In this paper we present new evidence on the predictability of excess returns on common stocks for the SP 500 and the Dow Jones portfolios at annual, quarterly, and monthly frequencies. On the basis of the regression results we compute recursive predictions of excess returns. Applying a non-parametric predictive failure test we show that these recursive predictions are capable of correctly forecasting a statistically significant proportion of the signs of the realized excess returns at all three trading horizons.
We show that the quarterly and annual switching portfolios constructed on the basis of the recursive predictions mean-variance dominate the respective market portfolios, paying higher mean returns with substantially lower standard deviations than the market, even under a high transaction cost scenario. At the monthly frequency the switching portfolios only mean- variance dominate the SP 500 market index when transaction costs are either zero or low. Our results also indicate that it may not pay off to trade as frequently as at the monthly interval unless transaction costs can be kept very low.
24 In order to derive Var(z) we have made use of the following formulae:
Var(z) = Var [E(z 1 x,y)l + E[Var(z 1 x,y) ]
E(zIx,y)=Pox+(1 -Po)Y V(Z I X7.Y) = Po( l - PO)(X - Y ) 2
360 Journal of Forecasting Vol. 13, Zss. No. 4
POSTCRIPT ’’ Here we provide new estimates of the excess return regressions given in Tables 1-111 using the additional annual, quarterly, and monthly observations over the period 1991 and 1992 that have become available since the first version of this paper was completed and submitted for publication. The new excess return regressions estimated over the longer period 1954-92 are summarized in Tables PI, PI1 and PI11 for annual, quarterly and monthly frequencies respectively. 26 The results in Tables PI-PI11 are directly comparable with those in Tables
Table PI. Postscript annual excess return regressionsa (sample period 1954-93)
SP 500 portfolio
b i z -0.274+ 8.91YSPi-I - 1.7OPIi-2-0.0660131-1 +O.IITERM~-I (0.073) (1.95) (0.43) (0.021) (0.038)
Z5This postscript was written in February 1993 on the suggestion of the editor. 26Note that the annual regressions are estimated over the period from January 1954 to January 1993.
M. Hashem Pesaran and Allan Timmermann Forecasting Stock Returns 361
DOW Jones xb~(2) = 0.86 Xb~(8) = 2.03 ~ b ~ ( 2 4 ) = 15.19
[0.704] [0.879] [0.811]
[0.649] [O. 9801 l0.9151
'The estimation periods for the predictive failure tests are the same as those in Tables 1-111. The sample period for the annual predictions is 1992-3, for the quarterly predictions it is 1991(1)-1992(4), and for the monthly predictions it is 1991(1)-1992(12). The statistics reported are the chi-squared version of the predictive failure test statistic described, for example, in Pesaran et a/. (1985, Section III), with the degrees of freedom given in parentheses. The figures in square brackets are the associated rejection probabilities.
Table PV. Recursive predictions and actual values of annual excess returns'
~~
SP 500 portfolio Dow Jones portfolio Years Actual Predicted Actual Predicted
M. Hashem Pesaran and Allan Timmermann Forecasting Stock Returns 363
1-111, and clearly show that the addition of the two extra years of data has had little effect on the parameter estimates and the diagnostic test statistics. Also using the regression results in Tables 1-111 to predict excess returns over the 1991-2 period do not show any statistically significant evidence of predictive failure. (See Table PIV). The annual, quarterly, and monthly recursive predictions of excess returns are given in Tables PV-PVII. These results reflect the favourable conditions that, under declining inflation and interest rates, have prevailed in the US stock market over the past two years. With the exception of the prediction for the Dow Jones portfolio over the period January 1992 to January 1993, which is only slightly negative, all other recursive predictions have turned out to be positive up to three decimal places. This is generally in line with the market outcomes. These predictions being positive do not, however, allow us to do better than the market over this particular period.
DATA APPENDIX
The annual observations are organized around the trading month, which is January. The dividend and price data are based on the SP 500 and Dow Jones Industrial portfolio. ‘Jan’ at the end of a variable name indicates that it is measured in January. ‘Oct’ that it is measured in October, and ‘Dec’ that it is measured in December. It is assdmed that trading takes place at closing time on the last trading day of January.
The timing of the variables is as follows: stock prices and interest rates on the last day of January are known at the time of trading. Dividends paid during the preceding year are also known at the end of January. The producer price index as well as other variables measured on a monthly basis are published with a lag of typically 15-20 days, such that the index for December in the preceding year is the latest available information at the time of trading.
In the quarterly and monthly cases trading takes place at closing time on the last trading day of each quarterlmonth. The timing of the variables is as follows. Stock prices and interest rates at the end of the quarter/month as well as dividends paid during the quarter/month are known at the closing time of the last trading day. Due to the delay in the publication of monthly measured macro variables, these will only be publicly available in the middle of the subsequent month.
Variables ZNPT PSP
DZVSP
NRSP
ERSP
YSP
Represents the intercept term in the regression equations. Nominal Price index for the SP 500 portfolio at the close of the last trading day of January (annual case) or the quarterlmonth. Source: Standard & Poor’s, Statistical Service. Average nominal dividends per share for the SP 500 portfolio paid during the calendar year. Source: Standard & Poor’s, Statistical Service. Nominal returns on the SP 500 portfolio, computed as NRSPt = (PSPt - PSP+ 1 + DZVSPt- I ) / PSPt- 1 (annual) NRSPr = (PSPt - PSPt- 1 + DZVSPt)/PSPt- 1 (quarterly/monthly). Excess returns on the SP 500 portfolio, computed as ERSPt = NRSPt - Zt-1, where It-1 = Z12Jant-l, Z3t-1, and Z l t - l in the case of annual, quarterly, and monthly frequencies, respectively. Dividend yield on the SP 500 portfolio, computed as YSPt = DZVSPt-l/PSPt (annual) YSPt = DZVSPt/PSPt (quarterly/monthly).
364 Journal of Forecasting Vol. 13, Iss. No. 4
PDJ
DIVDJ
NRDJ
ERDJ
PPIA V
PI IPA V DIP
BUS DBUS LEAD
DLEAD COIN
DCOIN LAGG
DLAGG I1
I3
I6
I12
Nominal Price index for the Dow Jones portfolio at the close of the last trading day of January (annual case) or the quarterlmonth. Source: The Dow Jones Investor’s Handbook 1991. Average nominal dividends per share for the Dow Jones portfolio paid during the preceding year. At the quarterly and monthly intervals, dividends are computed as 1/4 and 1/12 of the average annual dividends. Source: The Dow Jones Investor’s Handbook 1991. Nominal returns on the Dow Jones portfolio, computed as NRDJ, = (PDJ, - PDJ,-I + DIVDJt-l)/PDJPt-I (annual) NRDJ, = (PDJ, - PDJ,- + DIVDJ,)/PDJ,- 1 (quarterly/monthly). Excess returns on the SP 500 portfolio, computed as ERDJt = NRDJ, - &-I , where I , = I12Janr-1, 13t, and I l , in the case of annual, quarterly, and monthly frequencies. Annual average of Producer Price Index (PPI, finished goods). Source: Citibase (PWf). The 12-month inflation rate is computed as log(PPIA Vt/ PPIAV,- I ) .
12-month average of the industrial production index (IP). Source: Citibase (ip). The 12-month rate of change in industrial production computed as
12-month average of an index of net business formation. Source: Citibase (bus). Change in the business formation index, computed as DBUS, = log(BVS,/BUS,- 12).
12-month average of a composite index of 11 leading business cycle indicators. Source: Citibase (dlead). Change in the leading index, computed as DLEAD, = log(LEADr/LEADt- 12).
12-month average of a composite index of four roughly coincident business cycle indicators (seasonally adjusted). Source Citibase (dcoinc). Change in the coinciding index, computed as DCOINt = log(COINt/COZN1- 1 2 ) .
12-month average of a composite index of lagging business cycle indicators. Source: Citibase (dlagg). Change in the lagging index, computed as DLAGGI = log(LAGGr/LAGGr- 1 2 ) .
1-month T-bill rate on the last trading day of the month, annualized average of bid and ask yields. Source: CRSP tapes, the Fama-Bliss risk free ratesfrle. 3-month T-bill rate on the last trading day of the month, annualized average of bid and ask yields. Source: CRSP tapes, the Fama-Bliss risk free rates file. 6-month commercial paper rate, monthly average of daily figures. Source: Citibase (fYCP). 12-month discount bond rate on the last trading day of the month. Source: CRSP tapes, the Fama-Bliss discount bonds j l e . DIl l = 11-11 (- l), used in monthly regressions. DIl l2 = 11-11 ( - 12), used in quarterly regressions. D13 = I3 JAN-I3 OCT (- l), used in annual regressions. TERM = I6 JAN-I3 JAN, used in annual regressions.
log( IPA Vt/ IPA Vt - I ) .
Data used to predict excess returns over the period 1991-92 The Data sources were identical to the ones used over the period 1954-90 with the following exceptions. For the period January 1992 to January 1993, the 1-, 3-, and 12-month interest rates were downloaded from Datastream. In the case of stock prices and dividends for the
M. Hashem Pesaran and Allan Timmermann Forecasting Stock Returns 365
Dow Jones index we used Datastream for the period January 1991 to January 1993. DataStream was also used to get dividends and stock prices for the SP 500 portfolio over the period October 1992 to January 1993.
ACKNOWLEDGEMENTS
Helpful comments and suggestions by Pierluigi Balduzi, Tim Opler, Sean Holly, two anonymous referees of this journal, and by seminar participants at the London Business School, University of Southern California, the University of California at Riverside, Queen Mary College, London, and at the universities of Essex, Bristol, and Rome are gratefully acknowledged. The first author gratefully acknowledges financial support from the ESRC and the Newton Trust of Trinity College, Cambridge. The second author gratefully acknowledges financial support from the Carlsberg Foundation and the Danish Research Academy.
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M. Hashem Pesaran and Allan Timmermann Forecasting Stock Returns 361
A uihors ’ biographies: M. Hashem Pesaran is Professor of Economics at the University of Cambridge, and a Professorial Fellow of Trinity College, Cambridge. Previously he has been the head of the Economic Research Department of the Central Bank of Iran and the Under-Secretary of the Ministry of Education, Iran. Dr Pesaran is the founding editor of the Journal of Applied Econometrics. He has taught at Harvard University, UCLA, the University of Pennsylvania, and the Australian National University, among others. He has published widely in the areas of econometrics, empirical macroeconomics, and on the Iranian economy in leading economic and econometric journals. He is the author of several books and edited volumes, and is co-developer of the econometric software package Micro& 3.0, pubished by Oxford University Press. He is a Fellow of the Econometric Society, and a recipient of the Royal Economic Society Prize for the best article in the Economic Journal.
Allan Timmermann is currently a lecturer in economics at Birkbeck College, University of London, and a research affiliate of CEPR. He obtained MSc degrees in economics from the University of Copenhagen and the London School of Economics and holds a Ph.D in financial economics from the University of Cambridge. His research publications are in the areas of forecasting stock returns, trading in financial markets and how agents’ learning affects the dynamics of stock returns.
Authors ’ addresses: M. Hashem Pesaran, Faculty of Economics and Politics, University of Cambridge, Sidgwick Avenue, Cambridge, CB3 9DE.
Allan Timmermann, Birkbeck College, Gresse Street, London WlP 1PA
