1
Index Returns and Macroeconomic Variables in the United States.
Abstract
Due to the growth in popularity of ETF’s, models explaining excess returns on an index level have become increasingly important. The aim of this paper is to rekindle interest in one of these models, the Burmeister and Wall (1986) model, by examining the relationship between their four macroeconomic variables and index and sector-specific returns. I find that although the model is significant, only the default premium is priced. Moreover, the model does not withstand close scrutiny as both the significance of the model and the factor loadings are unstable. Therefore, it appears that the four macroeconomic variables are no longer consistently priced in the United States of America and thus the model should not be used to predict excess returns on indices.
Keywords: Asset Pricing, Macroeconomics and Risk Premia. JEL-codes: E44, G12.
Studentnr: s1916688 Name: Tom Koster
2
Introduction
In the past years, Exchange Traded Funds (ETF’s) have gained significant popularity amongst investors. It has therefore become increasingly important for investors to predict returns on for example indices. Because indices reflect the state of the economy, investors could benefit from a model that explains excess returns on indices by using macroeconomic variables.
Since the publication of Markowitz’ paper on modern portfolio theory (1952), investors have tried to forecast returns on stocks and portfolios. Several theories have been developed in order to achieve these results, including the two major ones, the Capital Asset Pricing Model (CAPM) and the Arbitrage Pricing Theory (APT). Whereas the CAPM is a single index model that is frequently used to explain stock returns, the APT is a multi-index model that can describe returns on both stocks and indices. Both models are used frequently, however, they are also criticised because of their lack of economic foundation. Burmeister and Wall (1986) propose an alternative model, which is derived from economic theory and is based on the APT; it uses unexpected changes in macroeconomic variables to explain excess returns on assets. Burmeister and Wall maintain that unexpected changes in the risk premium, unanticipated changes in the term structure of interest rates, unexpected changes in the rate of inflation and unanticipated growth in sales explain excess returns on the S&P500 and the T. Rowe Price New Horizons Fund, whereas Berry et al. (1988) argue that these variables also explain the excess returns on seven sectors, implying that indices do reflect the state of the economy. Although these results are interesting, the model has received little attention during the past decades.
3 S&P500, the T. Rowe Price New Horizons Fund and each of the six sectors. However, within the model, only the default premium is priced consistently for each of the aforementioned asset returns. Moreover, with respect to the second hypothesis, I find that when the period is divided up into four 72-month spans, the default premium is only weakly significant in the third period and strongly so in the fourth for the regression models explaining the excess returns on the S&P500. Moreover, when the time span for the models explaining the excess sector returns are divided up, the default premium is only consistently significant in the last 75-month period. Furthermore, it appears that the significance of the risk premium is driven by outliers during the period October 2008 - March 2009; when dummy variables are introduced, none of the macro-economic variables are significant. Finally, the model itself is significant during each of the periods when explaining the excess returns on the S&P500, however, the regression model is only significant during the third period when explaining excess sector returns.
This paper adds to existing literature by finding proof that the Burmeister and Wall model, although intuitively great, cannot be used for explaining excess returns on the S&P500 or the six S&P500 sectors. It can be concluded that even though there is demand for a model that can explain the returns on ETF’s, the Burmeister and Wall model is probably not the one.
The following section presents a review of existing literature in order to lay the groundwork for the paper. The literature review is followed by a section on the methodology and data used in this paper. Next are the results, where the outcomes of the research are presented and discussed. The penultimate section investigates the robustness of the research, whereas the final one provides the conclusions with respect to the paper.
Literature review
Since the publication of Markowitz’ paper on modern portfolio theory (1952), the field of finance has become highly technical. Markowitz proposed a basis for portfolio decisions of individual investors, he did not, however, provide a pricing model. Nonetheless, because of Markowitz’ ground-breaking work, two highly influential models have been proposed over the years.
4 controversy; although most researchers acknowledge the proposed positive relationship between risk and return, little support of the model is found, since a significant amount of the variation in stock returns cannot be explained by the market β (for example Basu (1977), Banz (1981), Bhandari (1988) and Rosenberg et al. (1985)).
Second, Ross (1976) proposed the Arbitrage Pricing Theory of capital assets pricing (APT). It differs from the CAPM in the fact that it is a multi-index model, in contrast to the single-index model that is the CAPM; i.e. the return generating process of the CAPM is based on the market portfolio and a risk free asset, while the APT tries to explain the excess returns of a stock or portfolio based on the sensitivities to several, a priori undefined factors through factor analysis, although the market portfolio is generally not one of these factors (Roll and Ross, 1980).
Moreover, whereas the CAPM relies on the efficient market portfolio, which is held by all investors, the APT assumes that there are no arbitrage options in the market. This implies that two portfolios with equal sensitivities to factors cannot have different returns, since this raises the possibility for an investor to sell the portfolio with the lower return and to hold a long position in the higher return portfolio for a riskless, positive expected return. The assumptions of immediate availability of the proceeds of short selling and the possibility of short selling altogether underlie the no arbitrage assumption. However, although these requirements are somewhat of a shortcoming of the APT, it has a great advantage in being a multi-index model, because the factors can be chosen with respect to the preferences of the practitioner. The assumptions that underlie the arbitrage pricing theory are not its only drawbacks, since the testability of the model itself has been discussed throughout previous literature (for example Shanken (1982) and Dybvig and Ross (1985)). Moreover, the factors that encompass the returns generating process function are not known with respect to the number of factors and, furthermore, the characteristics of these factors are unidentified. Several studies attempt to answer the question with respect to the number of factors (for example Langetieg, 1978). Roll and Ross (1980) maintained that “at least three factors are important for pricing, but that it is unlikely that more than four are present”.
Chen et al. (1986) propose that these four factors are “the spread between long and short interest rate, expected and unexpected inflation, industrial production and the spread between high- and low-grade bonds”. The choice for the aforementioned four factors can explained by reviewing the dividend discount model. If we write the value of a stock as:
𝑃𝑖 = ∑
𝐸(𝐷𝑖,𝑡)
1+𝑟𝑖,𝑡 𝑇
5 where 𝑃𝑖 is the price of stock 𝑖, 𝐸(𝐷𝑖,𝑡) the expected dividend of stock 𝑖 at time 𝑡 and 𝑟𝑖 the discount
rate of stock 𝑖 at time 𝑡, then it can be asserted that variables that affect either the numerator, the denominator or both, influence the value of the stock. First, since the discount factor is an average of interest rates over time, Chen et al. (1986) argue that it will be affected by both the level and the term structure of interest rates. Therefore, if the risk free rate changes, the asset pricing will be affected, as well as the asset returns, since the change will influence the time value of cash flows. Second, the risk premium affects the discount factor, consequently, unanticipated change in the premium influence asset pricing. Moreover, the risk premium is affected by changes in indirect marginal utility, hence it will also influence pricing. Third, the expected dividend is affected by unanticipated changes in industrial production, since this generally influences the level of cash flows. Fourth, inflation is related to both the numerator and the denominator of the model, it affects the expected cash flows and interest rate in nominal returns, as well as relative prices, since the change can affect the average inflation rate, which in its turn changes the valuation. This implies the following model for an index: 𝑅𝑖,𝑡 = 𝑎 + 𝑏𝑀𝑃𝑀𝑃𝑡+ 𝑏𝐷𝐸𝐼𝐷𝐸𝐼𝑡+ 𝑏𝑈𝐼𝑈𝐼𝑡+ 𝑏𝑈𝑃𝑅𝑈𝑃𝑅𝑡+ 𝑏𝑈𝑇𝑆𝑈𝑇𝑆𝑡+ 𝑒𝑖,𝑡, (2)
where 𝑅𝑖,𝑡 is the return of index 𝑖 at time 𝑡, 𝑎 a constant, 𝑀𝑃𝑡 the monthly change in industrial
production at time 𝑡, 𝐷𝐸𝐼𝑡 the change in expected inflation at time 𝑡, 𝑈𝐼𝑡 the unexpected inflation at
time 𝑡, 𝑈𝑃𝑅𝑡 the spread between high- and low-grade bonds at time 𝑡, 𝑈𝑇𝑆𝑡 the spread between long
and short interest rates at time 𝑡, the 𝑏𝑖 the factor loadings of the model and 𝑒𝑖,𝑡 the error term. The
model is estimated by first using factor analysis and afterwards replacing these factors by the aforementioned macroeconomic variables.
6 It is of utmost importance to understand what qualities the macroeconomic variables must possess in order to be appropriate risk factors. Berry et al. (1988) argue that the variables should be so that the factor is completely unpredictable at the start of the period, the factor has an extensive effect on stock returns and the factor must be priced, i.e. it should affect expected returns. First, the rationale for the first property of macroeconomic variables is the fact that if the factor is not unpredictable at the beginning of a period, investors want to be compensated for their exposure to the factor. Therefore, it is key that the expected value at the beginning of the period equals zero. An example from Berry et al. (1988) is the rate of inflation, which can be partially predicted and is therefore not an appropriate factor. However, the discrepancy between the predicted value of inflation and the actual value is appropriate, since this constitutes the unpredicted part. Second, the factor should have an extensive effect on stock returns, since this implies that the factor does not have a firm-specific effect. Third, an appropriate factor should have a non-zero price, because otherwise it would not have shown up in a factor analysis that estimates an APT function.
Subsequently, the relationship between unexpected changes in macroeconomic variables and returns has been studied frequently since the emergence of the CAPM and APT. For example, with a method based on the one used in Chen et al. (1986), Ewing et al. (2001a) found that shocks in real output, the stance in monetary policy and risk premia had significant impact on the returns of the S&P500. Moreover, Ewing et al. (2003) found that their method produced robust results when investigating the relationship between these variables and “five major S&P500 sector-specific stock market indices”. Based on Chen et al. (1986), Tsuji (2007) studied the relationship between Japanese stocks and the four macroeconomic variables as in (2), as well as some additional variables. However, Tsuji (2007) follows McElroy and Burmeister (1988) in estimating the coefficients, that is, instead of using factor analysis as in Chen et al. (1986) and replacing the factors with variables, Tsuji (2007) estimates the sensitivities and risk measures jointly. Tsuji finds that in Japan, not all Chen et al. (1986) factors are priced, rather, it mentions that innovations in money supply and those in gold and foreign exchange reserves are. Furthermore, Tsuji (2007) finds that the significance of for example the risk premium factor holds for the periods 1986 - 1991 and 1992 - 1997, however, for the period 1998 - 2003, the coefficient is not significant.
7 maintained that for North America, Europe and Japan, local models perform better than international ones. Furthermore, macroeconomic variables have been included in the Fama and French model (for instane Kim et al., 2011).
On the other hand, the CAPM has been transformed to include macroeconomic variables. Several authors contributed to the consumption capital asset pricing model (CCAPM), the relevance of which is determined by the assumption that changes in macroeconomic variables determine behaviour in stock markets (Chen, 2008). Although it performs poor as an asset pricing model (for example Cochrane, 1996), it has received a lot of attention because of its intuitive appeal. Moreover, Kang et al. (2011) maintain that their adaptation of the CCAPM, which includes a conditional variable, performs as well as the Fama and French three factor model. Furthermore, Chen (2008) asserts that “yield curve spreads and inflation rates are the most useful predictors of recessions in the US stock market”, implying that the model has value. It appears, however, that most of the more recent adaptations of both the CAPM and APT use different macroeconomic variables than the ones provided by Chen et al. (1986). For example, Chen (2008) includes dividend yield, Ewing et al. (2001) include monetary policy and innovation in FED funds rates and Tsuji (2007) includes innovations in money supply and gold and foreign reserves.
8 Horizons Fund, whereas Berry et al. (1988) found similar results for seven sectors. The relationships of each of the macroeconomic variables and returns are explored below.
Furthermore, as Panetta (2001) maintains, very little research has been done with respect to the stability of the factor loadings in arbitrage pricing models. Panetta finds that for Italian stocks, the coefficients in the factor regression are highly unstable. Even though Panetta uses factor analysis instead of selecting variables on a prior beliefs, the results are interesting and relevant for the Burmeister and Wall (1986) model.
This paper adds to existing literature by investigating the relationship by the four aforementioned macroeconomic variables introduced by Burmeister and Wall (1986) and excess returns on the S&P500 in a more contemporary period of time, as well as by checking the stability of variables in the model. By investigating the relationship from the period January 1990 - December 2013, it should become clear whether or not the Burmeister and Wall model is still relevant. Moreover, the significance of the relationship with excess returns on several sectors is explored, thus replicating Berry et al. (1988). Moreover, since the stability of the factor loadings in the Burmeister and Wall model has received little attention with respect to previous research, this paper could add new insights.
Methodology
As mentioned above, the aim of this paper is to establish whether or not the research of Burmeister and Wall (1986), as well as the subsequent research by Berry et al. (1988) is still relevant. Moreover, the stability of the factors is analyse, by reviewing the significance, as well as the absolute and relative size of the coefficients. First of all, then, the economic variables that contribute to the model have to be considered. The model of Burmeister and Wall is:
𝑅𝑖,𝑡− 𝑅𝑓,𝑡 = 𝛼 + 𝛽𝑖,𝑈𝑃𝑅∗ 𝑈𝑃𝑅𝑡+ 𝛽𝑖,𝑈𝑇𝑆∗ 𝑈𝑇𝑆𝑡+ 𝛽𝑖,𝑈𝐼∗ 𝑈𝐼𝑡+ 𝛽𝑖,𝑈𝑀𝑃∗ 𝑈𝑀𝑃𝑡+ 𝑒𝑖,𝑡, (3)
in which 𝑅𝑖,𝑡 represents the return on index or sector 𝑖 at time 𝑡, 𝑅𝑓,𝑡 the risk free return at time 𝑡,
𝑅𝑖,𝑡− 𝑅𝑓,𝑡 the excess return on an asset or portfolio of assets, including indices, 𝑈𝑃𝑅𝑡 the value of
unexpected change in the risk premium at time 𝑡, 𝑈𝑇𝑆𝑡 the value of unexpected change in the term
structure at time 𝑡, 𝑈𝐼𝑡 the value of unexpected change in the inflation rate at time 𝑡, 𝑈𝑀𝑃𝑡 the
unexpected growth in industrial production at time 𝑡, 𝛽𝑖,𝑘 the sensitivity to each of the macroeconomic
variables for an asset or portfolio of assets, including indices, where 𝑘 revers to the 𝑘-th variable in the equation, and 𝑒𝑖,𝑡 the error term.
9
𝑈𝑃𝑅𝑡 = 𝐶𝑌𝑡− 𝐿𝑇𝐺𝑌𝑡 , (4)
where 𝑈𝑃𝑅𝑡 represents the value of unexpected change in the risk premium at time 𝑡, 𝐶𝑌𝑡 the
holding-period yield on a portfolio of corporate bonds at time 𝑡 and 𝐿𝑇𝐺𝑌𝑡 the holding-period yield on a
portfolio of long term government bonds at time 𝑡. As mentioned above, an unexpected change in the risk premium could represent a change in risk aversion. Fama and French (1989) find that the default spread is negatively related to real returns. An explanation for this correlation could be that when “business conditions are poor, income is low and expected returns on bonds and stocks must be high to induce substitution from consumption to investment”.
Second, the unexpected change in term structure is estimated by:
𝑈𝑇𝑆𝑡= 𝐿𝑇𝐺𝑌𝑡− 𝑆𝑇𝐺𝑌𝑡−1, (5)
where 𝑈𝑇𝑆𝑡 is the value of unexpected change in the term structure at time 𝑡, 𝐿𝑇𝐺𝑌𝑡 the
holding-period yield on a portfolio of long term government bonds at time 𝑡 and 𝑆𝑇𝐺𝑌𝑡−1 the one-period
lagged yield on short term government bonds. The rationale for the lagged value is the fact that the yield on T-bills is first learnt at the end of the previous month, this is explained further below. Campbell (1987) finds that the term structure of interest rates can explain a significant portion of the excess returns and, moreover, the coefficients associated with the term structure were stable. Similar to the default spread, Fama and French (1989) assert that the term structure spread is negatively related to stock returns, because when business is doing well, the market clearing returns are at a lower level. Third, the unexpected change in rate of inflation is a somewhat more difficult variable to estimate with respect to the unexpected part of the variable. Since change in rate of inflation has to be unanticipated, it is key to forecast a value for that change. First then, the unexpected change in rate of inflation is:
𝑈𝐼𝑡 = 𝐼𝑡−1− 𝐸(𝐼𝑡−1), (6)
where 𝑈𝐼𝑡 is the value of unexpected change in the inflation rate at time 𝑡, 𝐼𝑡−1 the logarithmic change
in US Consumer Price Index at time 𝑡 − 1 and 𝐸(𝐼𝑡−1) the expected value of 𝐼𝑡−1. The expected value
of 𝐼𝑡−1 is estimated by an ARIMA model;1 the model produced a one-step-ahead prediction of 𝐼𝑡−1.
The ARIMA forecasts are based on an out-of-sample estimation, which is explained further in the data section, consequently, the Kalman filter technique used by Burmeister et al. (1986) is ignored. The lagged value can be explained by the delay in publication of the data, this is described more extensively below. Research on the effect of the rate of inflation on stock returns has been divided, for example, Fama (1983) finds a negative correlation, whereas Solnik and Solnik (1997) find a positive relationship.
1 The ARIMA model used for estimating the unexpected change in rate of inflation is based on the Schwarz
10 Lin (2009) maintains that unanticipated inflation has positive short term effects on stock returns indeed, however, in the long run, the relationship is negative.
Fourth, the unexpected change in industrial production is used as a proxy for long term growth rate of profits in an economy. For this variable, I follow Chen et al. (1986) in using industrial production as a proxy, instead of Burmeister and Wall’s unexpected growth in real final sales. The factor, which measures innovation in industrial production, has been measured by
𝑀𝑃𝑡 = log 𝐼𝑃𝑡−1− log 𝐼𝑃𝑡−2 (7)
in previous literature, however, as Panetta (2001) asserts, this does not deal with the strong serial correlation between growth rates of industrial production and therefore, following Burmeister and Wall (1986), it cannot be hypothesised that the growth rate is unexpected. Consequently, the growth has to be measured by estimating the value of 𝐼𝑃𝑡, which implies that the variable is represented by:
𝑈𝑀𝑃𝑡 = log 𝐼𝑃𝑡−1− 𝛦(log 𝐼𝑃𝑡−1), (8)
where 𝑈𝑀𝑃𝑡 is the value of unexpected change in industrial production at time 𝑡, log 𝐼𝑃𝑡−1 is the
natural logarithm of the lagged level of industrial production, 𝐼𝑃𝑡, and 𝛦(log 𝐼𝑃𝑡−1) is the expectation
of this value. The lagged value of industrial production is used, since the publication of the values is delayed approximately a month; this is further explained below. The expected value of industrial production, 𝛦(log 𝐼𝑃𝑡−1), is estimated by an ARIMA model;2 the model produced a one-step-ahead
prediction for 𝐼𝑃𝑡. The ARIMA forecasts are based on an out-of-sample estimation, which is explained
further in the data section. Consequently, the Kalman filter technique utilised by Burmeister et al. (1986) is ignored. By using an ARIMA model, the effect serial correlation and predictability problems are mitigated, however, it ignores previous research (for example Bodo, Cividini and Signorini (1991) and Marchetti and Parigi (1998)), which maintains that factor models outperform ARIMA in estimating industrial production; Bulligan et al. (2009) for example assert that “selected indicators and factor-based models always perform significantly better than ARIMA models”. Marathe and Shawky (1994) find that industrial production can significantly explain stock returns, moreover, the relationship is negative. Marathe and Shawky explain the sign by following the “Life Cycle Permanent Income hypothesis” of Modigliani and Brumberg, which implies that people smooth their consumption, by saving during prosperous periods in order to compensate when times get rough. Similarly, Marathe and Shawky argue that investment should be low during periods of low income, since income is being consumed instead of saved or invested; because of the lack of investment, output would be low.
2 The ARIMA model used for estimating the unexpected change in industrial production is based on the
11 Subsequently, with all the variables estimated, a time series regression is run on (3) to estimate the sensitivities, 𝛽𝑖,𝑘, to each of the variables. The relevance of the factors in the model can then quite
straightforwardly be measured by obtaining the probabilities associated with the 𝑡-values for each of the variables. Moreover, a 𝑓-test is performed in order to establish whether or not the model itself is relevant, i.e. to measure whether or not the factor loadings differ from zero. Furthermore, the time series regressions are not only run on markets, the effects of the macroeconomic variables on different sectors are considered as well, similar to Berry et al. (1988). In this paper, Berry et al. produce a ranking of each of the considered sectors with respect to each of the variables, a similar list is produced here.
Data and descriptive statistics
To assess whether or not the four aforementioned variables have a significant effect on excess returns, data requirements have to be fulfilled. In order to calculate the excess returns, monthly returns of the S&P500 and T. Rowe Price New Horizons Fund, as well as the yield of United States of America Treasury Bills, which is used as a proxy for the risk free rate, are retrieved from Thomson Reuters Datastream for the period January 1990 up to and including December 2013. The choice for these two specific returns follows from Burmeister and Wall (1986), since they determine the effect of the factors on both. To estimate the sensitivity of the sectors to each of the variables returns of the energy, financials, retail food, industrials, oil and gas and utilities sectors are obtained, where the sectors are constructed from stocks enlisted on the S&P500 by Thomson Reuters. The decision to examine these sectors is because they are similar to those explored in Berry et al. (1988). The sector data is obtained from Thomson Reuters Datastream, it consists of monthly returns from January 1995, the first available month, up to and including December 2013. Both the returns of the S&P500 and the T. Rowe Price New Horizons Fund, as well as the sector returns, have the advantage of being dynamic in nature, thus preventing the results being affected by the survivor bias.
With respect to the data required for the estimation of the variables, I first consider the unexpected change in risk premium, 𝑈𝑃𝑅𝑡. 𝐶𝑌𝑡, the yield on a portfolio of corporate bonds is obtained from
Thomson Reuters Datastream, considers the yield on corporate bonds rated AAA and entails the period January 1990 - December 2013. The variable, which has the Datastream-ticker “LHIGAAA”, is constructed by Barclays. 𝐿𝑇𝐺𝑌𝑡, the yield on long term government bonds is estimated by a proxy, the
United States of America Treasury yield with a maturity of 20 years, entailing a period corresponding to the one mentioned above for 𝐶𝑌𝑡. The proxy is obtained from Datastream and is characterised by
the ticker “USGBOND.”. Second, to determine the unexpected change in term structure, 𝑈𝑇𝑆𝑡, the
same long term government yield is utilised, as well as 𝑆𝑇𝐺𝑌𝑡−1, for which the one-period-lagged yield
12 thus implying a lagged value. The short term yield has an equal amount of data points to the previous datasets. Third, the inflation variable, 𝑈𝐼𝑡, is estimated by using monthly US Consumer Price Index (CPI)
data from the US Bureau of Labor Statistics,3 a dataset consisting of values from the period January 1980 - December 2013. The first ten years of the period, i.e. January 1980 - December 1989, is used for estimation purposes, thus implying an out-of-sample estimation as mentioned above. However, this only holds for the estimations for S&P500 and T. Rowe Price New Horizons Fund; for the sector returns the first fifteen years are used to forecast the out-of-sample values of the rate of inflation, since the sector returns are firstly available for January 1995. Since the Bureau of Labor Statistics generally publishes the CPI data during the third week after the end of the month it refers to, the data is lagged one month. Fourth, 𝑈𝑀𝑃𝑡 is estimated in a similar fashion as 𝑈𝐼𝑡 with respect to the periods
used for the out-of-sample forecasts. The industrial production variable is measured with data from the US Federal Reserve Bank,4 which generally publishes industrial production data in the third week after the end of the month it refers to, thus explaining the rationale for the lagged value.
The dataset results from the estimations as described earlier can be characterised as shown in Table 1. It follows from Table 1 that none of the macroeconomic variables is normally distributed. Moreover, the excess returns are far from normally distributed. Noteworthy is the fact that the excess returns are negative, which implies that both the S&P500 and the hedge fund provide returns south of the yield on three-month US T-bills, at a higher risk. Furthermore, the hedge fund does not outperform the market, in this case the S&P500. However, none of these results is that interesting without the context of the time series regression.
Table 2 depicts the correlations between the various macroeconomic variables as well as those with the excess returns on the S&P500. First, the relatively high correlations between excess returns on the S&P500 with respectively the risk premium factor and the term structure factor are noteworthy. It appears that an increase in in risk premium correlates with a lower return on the S&P500, which makes intuitively sense, since this would imply that interest payments on corporate bonds increase, making financing more expensive. Moreover, it appears that the return on the S&P500 and the term structure factor move together, which is somewhat more difficult to interpret and contradicting to the results found by Chen et al. (1986). Second, although the correlations are lower, the positive correlation between the S&P500 and 𝑈𝐼𝑡 implies that inflation moves together with increases in the dividends for
investors, whereas the correlation with the industrial production factor implies that growth in the
3 United States Consumer Price Index dataset can be retrieved from the US Bureau of Labor Statistics at
http://www.bls.gov/cpi/data.htm.
4 United States Industrial Production dataset can be retrieved from the US Federal Reserve Bank at
13 economy is accompanied by a negative effect on profits. The latter correlations are somewhat difficult to interpret and contradict the results of Chen et al. (1986). Lastly, the high, negative correlation between the risk premium factor and the term structure factor can most likely be explained by the fact that both are estimated by the yield on a portfolio of treasury bonds with a maturity of 20 years. Chen et al. (1986) find a similar correlation and assert that using different portfolios for either variable does not yield better results.
Descriptive UPR UTS UI UMP S&P500 T. Rowe Price
Mean 0.0004 0.0230 -0.0002 -0.0001 -0.0245 -0.0172
Std. Dev. 0.0066 0.0132 0.0028 0.0057 0.0495 0.1465
Skewness 0.7296 -0.1686 -0.6459 -0.6426 -0.1490 -1.4096
Kurtosis 5.3525 1.8379 6.3326 7.9724 3.7816 10.3817
Jarque-Bera 91.6401 17.5082 152.7603 315.4146 8.3664 746.6504
Table 1: The descriptive statistics of the four macroeconomic variables used to explain excess returns, as well as those of the excess returns on the S&P500 and the T. Rowe Price New Horizons Fund. Here UPR represents the variable unexpected change in the risk premium, UTS the variable unexpected change in the term structure, UI the variable unanticipated change in rate of inflation, UMP the unanticipated growth in industrial production, S&P500 the excess returns on the index over the yield on the T-bills and T. Rowe Price the excess returns of this fund. The descriptive statistics are generated for period January 1990 - December 2013 from the variables estimated by the datasets obtained from Thomson Reuters Datastream, the US Bureau of Labor Statistics and the US Federal Reserve Bank.
Name S&P500 UI UMP UPR UTS
S&P500 1
UI 0.0140 1
UMP -0.0691 -0.0829 1
UPR -0.4589 -0.1404 -0.1301 1
UTS 0.3221 -0.0091 0.0493 -0.5644 1
Table 2: The correlation matrix of the macroeconomic variables, as well as those with the excess returns on the S&P500. Here UPR represents the variable unexpected change in the risk premium, UTS the variable unexpected change in the term structure, UI the variable unanticipated change in rate of inflation, UMP the unanticipated growth in industrial production, S&P500 the excess returns on the index over the yield on the T-bills. The correlations are estimated for the period January 1990 - December 2013 from the variables estimated by the datasets obtained from Thomson Reuters Datastream, the US Bureau of Labor Statistics and the US Federal Reserve Bank.
Moreover, in Figures 1 - 4 in the appendix, the values of the macroeconomic variables throughout the years are displayed, as well as the respective excess returns on the S&P500. For example, Figure 1 depicts the situation with respect to 𝑈𝑃𝑅𝑡 and the excess returns. From Figure 1 follows that the values
of 𝑈𝑃𝑅𝑡 and the excess returns on the S&P500 move somewhat together, which was also mentioned
above. Moreover, it appears that in the six months following September 2008, the values of 𝑈𝑃𝑅𝑡
peak, most likely caused by the growing uncertainty in financial markets, that is, it appears the value of the default premium increased during this period.
Figure 2 depicts the values of 𝑈𝑇𝑆𝑡 and the excess returns on the S&P500 during the sample period. It
appears that the variable and the excess returns move together to some extent, although 𝑈𝑇𝑆𝑡
14 on the S&P500 and the values of 𝑈𝐼𝑡 and 𝑈𝑀𝑃𝑡 respectively are hard to interpret, the movements with
regard to each other are somewhat random, which was also demonstrated by the low correlations in Table 2. However, it is key to discuss the results in the context of a time series regression.
Results
The results that follow from the time series regression on the excess returns of the S&P500 are discussed in Table 3. First, it follows from Table 3 that the regression model is significant for the S&P500, which is characterised by the 𝑓-statistic of 21.7771. Moreover, the Durbin-Watson-statistics show that auto-correlation is not a problem within the model. Second, the high significance of the constant in the model implies that there is other market risk explaining excess returns beyond the four macroeconomic variables. With respect to the research of Burmeister and Wall (1986), the results in Table 3 oppose the coefficients found in the paper. Burmeister and Wall described positive coefficients for 𝑈𝑃𝑅𝑡, 𝑈𝑇𝑆𝑡 and 𝑈𝑀𝑃𝑡, whereas the results in Table 3 depict that only the term structure premium
is positive. However, the negative coefficient of 𝑈𝐼𝑡 corresponds with the results of Burmeister and
Wall. Most importantly, whereas in Burmeister and Wall all coefficients, excluding the constant, are significant, Table 3 displays that significance only holds, albeit weakly, for the profits growth variable and for the risk premium factor in the model. The results with respect to the T. Rowe Price New Horizons Fund can be found in table 6 in the appendix.
Berry et al. (1988) expanded on the research of Burmeister and Wall by discussing the sensitivities of sector excess returns to the macroeconomic variables; Table 4 displays these premia for a selected number of sectors. The relevance of their research follows from a portfolio perspective, investors can more easily choose their exposure with respect to each of the variables, by choosing a sector that has their preferred level of sensitivity. Moreover, investors holding a position in each of the sectors can hedge their exposure more readily if they understand the level of sensitivity. Although Table 4 discusses few different sectors than the Berry et al. paper, for example energy, industrials and retail food, it overlaps with respect to the financial, oil and gas and utility sectors. It follows from Table 4 that the constant is significant for all sectors excluding the financials; this implies that there is significant market risk beyond the four premia discussed here.
15
Name C UPR UTS UI UMP F-statistic DW Adj. 𝑹𝟐
S&P500 -0.0305** -3.2758** 0.3002 -1.0350 -1.1655* 21.7771** 1.8847 0.2252 (4.9822) (6.7902) (1.2641) (1.0935) (2.5568)
Table 3: The coefficients of the premia associated with each of the macroeconomic variables, as well as some descriptive statistics of the model; the F-statistic, Durbin-Watson-statistic and the adjusted r-squared. C represents the constant in the equation 𝑅𝑖,𝑡− 𝑅𝑓,𝑡= 𝛼 + 𝛽𝑖,𝑈𝑃𝑅∗ 𝑈𝑃𝑅𝑡+ 𝛽𝑖,𝑈𝑇𝑆∗ 𝑈𝑇𝑆𝑡+ 𝛽𝑖,𝑈𝐼∗ 𝑈𝐼𝑡+ 𝛽𝑖,𝑈𝑀𝑃∗ 𝑈𝑀𝑃𝑡, UPR the variable unexpected change in the risk premium, UTS the variable unexpected change in the term structure, UI the variable unanticipated change in rate of inflation, UMP the unanticipated growth in industrial production. The second and the fourth row in the table depict the coefficients in the equation; i.e. 𝛽𝑖,𝑘, whereas the bracketed values found in the third and fifth column represent the t-statistics associated with the coefficients. * implies that the coefficient or model is significant at a 5% threshold, whereas ** indicates that it is significant at a 1% level. The regressions are run on the variables calculated from the dataset retrieved from Thomson Reuters Datastream, US Bureau of Labor Statistics and the US Federal Reserve Bank for the period January 1990 - December 2013.
Name C UPR UTS UI UMP F-statistic DW Adj. 𝑹𝟐
Energy -0.0245** -3.0065** 0.2831 1.7538 -0.8097 10.3334** 2.2961 0.1418 (2.8428) (4.2618) (0.8095) (1.2928) (1.2173) {4} {3} {5} {6} {3} Financials -0.0159 -4.6284** -0.2898 0.6651 -1.0417 10.7637** 2.1086 0.1473 (1.5373) (5.4610) (0.6898) (0.4081) (1.3036) {6} {1} {1} {4} {1} Industrials -0.0240** -3.8503** 0.1306 -1.4799 -0.7396 14.9926** 1.9656 0.1985 (3.0470) (5.9742) (0.4089) (1.1940) (1.2172) {5} {2} {2} {1} {5}
Oil and Gas -0.0285** -2.3208** 0.4656 1.1786 -0.7975 9.3719** 2.3277 0.1291 (3.6471) (3.6224) (1.4662) (0.9566) (1.3201) {2} {4} {6} {5} {4} Food Retail -0.0302** -2.2744** 0.2373 -0.7612 -0.9511 5.2311** 1.9860 0.0697 (3.4516) (3.1703) (0.6673) (0.5517) (1.4060) {1} {5} {4} {3} {2} Utilities -0.0265** -2.1355** 0.1900 -0.7620 -0.5142 6.7784** 1.8618 0.0928 (3.8015) (3.7440) (0.6721) (0.6947) (0.9561) {3} {6} {3} {2} {6}
Table 4: Results from the time series regressions on the excess returns of the sectors, as well as some descriptive statistics of the model; the F-statistic, Durbin-Watson-statistic and the adjusted r-squared. C represents the constant in the equation 𝑅𝑖,𝑡− 𝑅𝑓,𝑡= 𝛼 + 𝛽𝑖,𝑈𝑃𝑅∗ 𝑈𝑃𝑅𝑡+ 𝛽𝑖,𝑈𝑇𝑆∗ 𝑈𝑇𝑆𝑡+ 𝛽𝑖,𝑈𝐼∗ 𝑈𝐼𝑡+ 𝛽𝑖,𝑈𝑀𝑃∗ 𝑈𝑀𝑃𝑡, UPR the variable unexpected change in the risk premium, UTS the variable unexpected change in the term structure, UI the variable unanticipated change in rate of inflation, UMP the unanticipated growth in industrial production. The first of the three rows of each sector depicts the coefficient associated with each of the variables in the model, i.e. 𝛽𝑖,𝑘. The second row displays the t-statistics measuring the significance of the coefficients, whereas the third row exhibits the ranks of the coefficients in within the six sector sample; rank 1 being the lowest number, 6 being the highest. * implies that the coefficient or model is significant at a 5% threshold, whereas ** indicates that it is significant at a 1% level. The regressions are run on the variables calculated from the dataset retrieved from Thomson Reuters Datastream, US Bureau of Labor Statistics and the US Federal Reserve Bank for the period January 1995 - December 2013.
16 Additionally, the insignificant coefficient of the energy sector associated with the inflation variable is noteworthy, it appears that energy companies cannot benefit from unanticipated increases in inflation, which can be explained by considering the relative robustness of their rates due to long term contracts. The insignificant premia for the utilities sector is in accordance with Berry et al. (1988); the sensitivity should not be significant, since the profits of this sector are highly regulated by the US government to keep the profit ratio stable. This implies that the effect of unexpected changes in the rate of inflation should be small, since the exposure is limited. The signs of the 𝑈𝐼𝑡 coefficients
correspond mostly with Berry et al., excluding the energy and financials sector. However, the insignificance of the coefficient for the financial sector does not; it appears that financials are more readily able to pass along the increases in costs and more able to hedge exposure than in the period January 1972 - December 1982.
The low, insignificant coefficient for 𝑈𝑀𝑃𝑡 with respect to the utilities sector coincides with Berry et
al.; due to the regulations on profits, the effect of unanticipated growth in industrial production, which is used as a proxy for unexpected changes in the growth rate of profits, is cushioned. It follows from Table 4 that none of the coefficients of 𝑈𝑀𝑃𝑡 is significant, which could imply that the sectors are more
diversified and therefore less exposed to changes in growth in profit rates. Finally, it is key to mention that the models are significant, even though this is not the case for the individual risk exposures. Obviously the most striking result displayed in Tables 3, 4 and 6 is the consistent significance of 𝑈𝑃𝑅𝑡;
the coefficients are significant for both the S&P500 and T. Rowe Price New Horizons Fund models, as well as for each of the sector models. It is therefore key to determine whether these results withstand closer scrutiny.
Robustness
It followed from the results section that within the Burmeister and Wall (1986) model the risk premium factor, 𝑈𝑃𝑅𝑡, is strongly significant for in the model explaining the excess returns on the S&P500, as
well as those models explaining the excess sector returns. Now, the stability of the results with respect to this factor are studied.
17 to and including the last month previous to the beginning of the period for an out-of-sample forecast; for example, the ARIMA model for the second period is estimated using the dataset for CPI and industrial production for the months January 1980 - December 1995. Similarly to the results reported above, time series regressions were used to estimate the risk exposures of the index and hedge fund with respect to the macroeconomic variables; the results for the S&P500 can be found in Table 5, whereas the results with respect to the T. Rowe Price New Horizons Fund are displayed in Table 7 in the appendix.
Name Period C UPR UTS UI UMP F-statistic DW Adj. 𝑹𝟐
S&P500 1 -0.1455** 0.5457 2.3019** -6.9176** -1.1117 10.6332** 1.6068 0.3550 2 -0.0874** -3.3430 0.8389 0.4569 -1.2890 4.0704** 2.1372 0.1493 3 -0.0689 -3.5385* 0.4749 -1.9268 -0.0687 12.5181** 2.1066 0.3969 4 -0.0494 -4.2245** 1.1397 -1.1071 -1.5927 7.6223** 1.9181 0.2745 Table 5: The coefficients of the premia associated with each of the macroeconomic variables, as well as some descriptive statistics of the model; the F-statistic, Durbin-Watson-statistic and the adjusted r-squared. C represents the constant in the equation 𝑅𝑖,𝑡− 𝑅𝑓,𝑡= 𝛼 + 𝛽𝑖,𝑈𝑃𝑅∗ 𝑈𝑃𝑅𝑡+ 𝛽𝑖,𝑈𝑇𝑆∗ 𝑈𝑇𝑆𝑡+ 𝛽𝑖,𝑈𝐼∗ 𝑈𝐼𝑡+ 𝛽𝑖,𝑈𝑀𝑃∗ 𝑈𝑀𝑃𝑡, UPR the variable unexpected change in the risk premium, UTS the variable unexpected change in the term structure, UI the variable unanticipated change in rate of inflation, UMP the unanticipated growth in industrial production. The values in columns 3-7 depict the coefficients in the equation; i.e. 𝛽𝑖,𝑘. * implies that the coefficient or model is significant at a 5% threshold, whereas ** indicates that it is significant at a 1% level. The regressions are run on the variables calculated from the dataset retrieved from Thomson Reuters Datastream, US Bureau of Labor Statistics and the US Federal Reserve Bank. The first period consists of the 72 months January 1990 - December 1995, where the data from January 1980 - December 1989 is used for estimation of IP and I by the ARIMA models. The second period consists of the 72 months January 1996 - December 2001, where the data from January 1980 - December 1995 is used for estimation of IP and I by the ARIMA models. The third period consists of the 72 months January 2002- December 2007, where the data from January 1980 - December 2001 is used for estimation of IP and I by the ARIMA models. The fourth period consists of the 72 months January 2008 - December 2013, where the data from January 1980 - December 2007 is used for estimation of IP and I by the ARIMA models. Each of the ARIMA models for IP and I in each of the periods is based on its individual Schwarz Information Criterion.
It follows from Table 5 that the coefficients, 𝛽𝑖,𝑘, are quite unstable. None of the risk exposures is
consistently significant, which implies that the variable is relevant in some periods, but is irrelevant in others. More specifically, it appears that although the risk premium variable, 𝑈𝑃𝑅𝑡, is weakly
significant in the third period and strongly so in the fourth, it is not significant for the first two. Moreover, the sign of the variable reverses between the first two periods. This result demonstrates the instability of the factor loadings. The model itself, however, is significant during each of the periods. Additionally, Table 8 in the appendix discusses the results of the stability analysis of the regression models explaining the excess returns on the sector portfolios. First, the data associated with the time series regressions performed above is divided in three periods, January 1995 - April 2001, May 2001 - September 2007 and October 2007 - December 2013 respectively. For each of the periods, 𝐼𝑡 and 𝐼𝑃𝑡
18 models are mostly insignificant in the earlier periods, whereas 𝑓-statistics depict that in the third period the regression models are significant for all sectors. Moreover, the signs and significance of the coefficients are inconsistent throughout the periods; however, with respect to the risk premium factor, 𝑈𝑃𝑅𝑡, the sign is relatively stable, it is consequently negative in each of the periods for each of the
sectors apart from the food retail sector during period 1, where the coefficient is 0.1487. Furthermore, with respect to the significance, in accordance with the results displayed in Table 5, 𝑈𝑃𝑅𝑡, is strongly
significant for each of the sectors in period 3, as well as strongly significant during period 1 for the energy sector and weakly so for the oil and gas and industrials sectors during the first and second period respectively. These results indicate that the model does not perform particularly well in explaining excess sector returns, apart from the third period, where both the model and 𝑈𝑃𝑅𝑡 are
significant. This could imply that the significant results are driven by some outliers in the third period. Furthermore, the ranks displayed in Table 8 show the consistency relative to the sample of sectors. This second check of the robustness of the results compares the ranks of each coefficient for each sector portfolio for each of the three periods. Table 8 displays that with respect to 𝑈𝑃𝑅𝑡, the financial
sector excess returns has the most negative coefficient in the first period, the sixth in the second and fifth during the last, whereas the coefficient associated with the retail food sector portfolio is ranked second, fourth and sixth for the first, second and third period respectively. In fact, the ranks are unstable for each of the variables to some degree.
Since the significance of the risk premium factor might be driven by outliers, the regression models are run including dummy variables for both the months October 2008 - March 2009, as well as one for the whole period. It appears that the results are indeed driven by the growing uncertainty in the financial markets during this period, since 𝑈𝑃𝑅𝑡 becomes insignificant in the regression model explaining the
excess returns on the S&P500 when monthly dummy variables are included. However, when a dummy variable for the whole period is included, the risk premium variable is significant on a 5% level. However, with respect to the excess sector returns, this problem does not exist, 𝑈𝑃𝑅𝑡 becomes
insignificant regardless of whether the dummies are monthly or whole period dummies. However, since the period concerns are six month time frame, it is questionable whether the usage of dummies is justifiable.
In summary, it appears that the risk exposures of indices, sectors and industries with respect to the four macroeconomic variables discussed in this paper, 𝑈𝑃𝑅𝑡, 𝑈𝑇𝑆𝑡, 𝑈𝐼𝑡 and 𝑈𝑀𝑃𝑡 are highly
inconsistent, since both the sign, as well as the absolute and relative size of the coefficients are unstable. It appears, however, that 𝑈𝑃𝑅𝑡, the unanticipated change in the risk premium, is consistently
19 period. In spite of the significance, the robustness analysis suggests that 𝑈𝑃𝑅𝑡 is not significant when
dummy variables are used for the months October 2008 - March 2009.
Conclusions
Despite combining the intuitive strength of the arbitrage pricing theory and the theoretical, macroeconomic foundation for the factors used in this APT models, the Burmeister and Wall (1986) model, which is based on the work of Chen, Roll and Ross (1986), has received little attention. Moreover, the model has not been tested with respect to the stability of the factor loadings.
This paper analysed the relationship between the macroeconomic variables proposed by Burmeister and Wall and the excess returns on the S&P500, T. Rowe Price New Horizons Fund, and six sectors, as well as the stability of the sensitivities to the macroeconomic variables. I find that only 𝑈𝑃𝑅𝑡, the
default premium, is significant for the aforementioned index, fund and sectors during January 1990 - December 2013 in the United States. However, it appears that instability of the sensitivities is a very serious problem; for each of the factor loadings on the macroeconomic variables, 𝑈𝑃𝑅𝑡, 𝑈𝑇𝑆𝑡, 𝑈𝐼𝑡 and
𝑈𝑀𝑃𝑡 it holds that both the signs, as well as the absolute and relative size differ between sub periods.
It can therefore be concluded that although the Burmeister and Wall model itself is often significant, the individual variables in the equations are not and consequently the model is not an appropriate tool for investors. This can for example be demonstrated by taking a look at hedging the exposure to one of the variables, where the position would never be truly hedged due to the instability of the exposure. Moreover, investors could use the model to estimate excess returns, however, these returns would be highly uncertain due to the instability and therefore not quite as useful as expected.
Overall, the findings of this paper are in contrast with Burmeister and Wall (1986), since it provides evidence that the macroeconomic variables in the model are not necessarily relevant. However, this results is in accordance with most of the more recent literature, such as Ewing et al. (1993) and Tsuji (2007), since it implies that the model and framework are relevant, but different macroeconomic variables should be used.
The main limitation of this paper is the fact that the Burmeister and Wall framework is used to produce static coefficients, which is unrealistic. It is more appropriate to using rolling regressions to estimates exposures and use these results to test the stability. However, this would defeat the purpose of the paper, which is to check the relevance of the model. Furthermore, Burmeister and Wall use the Kalman technique to determine the values of both 𝑈𝐼𝑡 and 𝑈𝑀𝑃𝑡, whereas this paper uses a simple ARIMA
20
References
Avramov, D., Chordia, T., (2006), Asset Pricing Models and Financial Market Anomalies, Review of Financial Studies, 19, 1001–1040.
Banz, R., (1981), The Relationship Between Return and Market Value of Common Stocks, Journal of Financial Economics. 9 (1), 3–18.
Basu, S., (1977), Investment Performance of Common Stocks in Relation to Their Price- Earnings Ratios: A Test of the Efficient Market Hypothesis, Journal of Finance, 12 (3), 129–156.
Bhandari, L., (1988), Debt/Equity Ratio and Expected Common Stock Returns: Empirical Evidence, Journal of Finance, 43 (2), 507–28.
Berry, M., Burmeister, E. and McElroy, M., (1988), Sorting Out Risking Using Known APT Factors, Financial Analysts Journal, (2), 29-41.
Bodo, G., Cividini, A. and Signorini, L., (1991), Forecasting the Italian Industrial Production Index in Real Time, Journal of Forecasting, 10, 285-99.
Bulligan, G., Golinelli, R. and Parigi, G., (2010), Forecasting Monthly Industrial Production in Real-Time: from Single Equations to Factor-Based Equations, Empirical Economics, 39, 303-336.
Burmeister, E., Wall, K., (1986), The Arbitrage Pricing Theory and Macroeconomic Factor Measures, The Financial Review, 21 (1), 1-20.
Burmeister, E., Wall, K. and Hamilton, J., (1986), Estimation of Unobserved Expected Monthly Inflation Using Kalman Filtering, Journal of Business and Economic Statistics, 4 (2), 147-160. Carhart, M., (1997), On Persistence in Mutual Fund Performance, Journal of Finance, 52, 57–82. Campbell, J., (1987), Stock Returns and the Term Structure, Journal of Financial Economics, 18, 373– 399.
Chen, N., Roll, R. and Ross, S. (1986), Economic Forces and the Stock Market, Journal of Business, 59 (3), 383–403.
Chen, S., (2009), Predicting the Bear Stock Market: Macroeconomic Variables as Leading Indicators, Journal of Banking and Finance, 33, 211-223.
Cochrane, J., (1996), A Cross-sectional Test of an Investment-based Asset Pricing Model, Journal of Political Economy, 104, 572–621.
Dybvig, P., Ross, R., (1985), Yes, the APT is Testable, Journal of Finance, 40 (4), 1173-1188.
Ewing, B., (2001), Cross Effects of Fundamental State Variables, Journal of Macroeconomics, 23, 633-645.
Ewing, B., Forbes, S. and Payne J., (2003), The Effect of Macroeconomic Shocks on Sector-Specific Returns, Applied Economics, 35, 201-207.
21 Fama, E., French, K., (1989), Business Conditions and Expected Returns on Stocks and Bonds, Journal of Financial Economics, 25, 23–49.
Fama, E., French, K., (1993), Common Risk Factors in the Returns on Stocks and Bonds, Journal of Financial Economics, 33, 3–56.
Fama E., French, K., (2012), Size, Value, and Momentum in International Stock Returns, Journal of Financial Economics, 105 (3), 457.
Griffin J., (2002), Are the Fama and French factors global or country specific?, Review of Financial Studies, 15, 783–803.
Kang, J., Kim, T., Lee, C. and Min B., (2011), Macroeconomic Risk and the Cross-section of Stock Returns, Journal of Banking and Finance, 35, 3158-3173.
Kim, D., Kim, T. and Min, B., (2011), Future Labor Income Growth and the Cross-section of Equity Returns, Journal of Banking and Finance, 35 (1), 67-81.
Langetieg, T., (1978), An Application of a Three-Factor Performance Index to Measure Stockholder Gains from Merger, Journal of Financial Economics, 6, 365-383.
Lin, S., (2009), Inflation and Real Stock Returns Revisited, Economic Enquiry, 47 (4), 783-795.
Lintner, J. (1965), Security Prices, Risk and Maximal Gains from Diversification, Journal of Finance, 20 (4), 587-615.
Marathe, A., Shawky, H., (1994), Predictability of Stock Returns and Real Output, Quarterly Review of Economics and Finance, 34 (4), 317-331.
Marchetti, D., Parigi, G., (1998), Energy Consumption, Survey Data and the Prediction of Industrial Production in Italy, Banca d’Italia, Temi di Discussione, 342.
Markowitz, H. (1952), Portfolio Selection, Journal of Finance, 7 (1), 77-91.
McElroy, M., Burmeister, E., (1988), Joint Estimation of Factor Sensitivities and Risk Premia for the Arbitrage Pricing Theory, Journal of Finance, 43 (3), 721-733.
Mossin, J. (1966), Equilibrium in a Capital Asset Market, Econometrica, 34 (4), 768-783.
Panetta, F., (2001), The Stability of the Relation between the Stock Market and Macroeconomic Factors, Temi di discussione del Servizio Studi, 393.
Rapach, D., Wohar, M. and Rangvid, J., (2005), Macro Variables and International Stock Return Predictability, International Journal of Forecasting, 21 (1), 137-166.
Roll, R., Ross S., (1980), An Empirical Investigation of the Arbitrage Pricing Theory, Journal of Finance, 35 (5), 1073-1103.
Roll, R., Ross S., (1995), The Arbitrage Pricing Theory Approach to Strategic Portfolio Planning, Financial Analysts Journal, (1), 122-131.
22 Ross, S., (1976), The Arbitrage Theory of Capital Asset Pricing, Journal of Economic Theory, 13 (6), 341-360.
Shanken, J. (1982), The Arbitrage Pricing Theory: Is it Testable?, Journal of Finance, 37 (5), 1129-1140. Sharpe, W., (1964), Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk, Journal of Finance, 19 (3), 425-442.
Solnik, B., Solnik, V., (1997), A Multi-Country Test of the Fisher Model for Stock Returns, Journal of International Financial Markets Institutions and Money, 7, 289–301.
23
Appendix
Figure 1: The values of UPR and the excess returns on the S&P500 throughout the timespan of the study. On the left Y-axis the values of UPR are displayed, whereas those of the excess returns on the S&P500 can be found on the right Y-axis. The values of UPR follow from UPRt= CYt− LTGYt, where UPRt represents the value of unexpected change in the risk premium at time t, CYt the holding-period yield on a portfolio of corporate bonds at time t and LTGYt the holding-period yield on a portfolio of long term government bonds at time t. The values are generated for period January 1990 - December 2013 from the variables estimated by the datasets obtained from Thomson Reuters Datastream, the US Bureau of Labor Statistics and the US Federal Reserve Bank.
24 Figure 3: The values of UI and the excess returns on the S&P500 throughout the timespan of the study. On the left Y-axis the values of UI are displayed, whereas those of the excess returns on the S&P500 can be found on the right Y-axis. The values of UI follow from UIt= It−1− E(It−1), where UIt is the value of unexpected change in the inflation rate at time t, It−1 the logarithmic change in US Consumer Price Index at time t − 1 and E(It−1) the expected value of It−1. The values are generated for period January 1990 - December 2013 from the variables estimated by the datasets obtained from Thomson Reuters Datastream, the US Bureau of Labor Statistics and the US Federal Reserve Bank.
Figure 4: The values of UMP and the excess returns on the S&P500 throughout the timespan of the study. On the left Y-axis the values of UMP are displayed, whereas those of the excess returns on the S&P500 can be found on the right Y-axis. The values of UMP follow from UMPt= 𝑙𝑜𝑔 IPt−1− Ε(𝑙𝑜𝑔 IPt−1), where UMPt is the value of unexpected change in industrial production at time t, 𝑙𝑜𝑔 IPt−1 is the natural logarithm of the lagged level of industrial production, IPt, and Ε(𝑙𝑜𝑔 IPt−1) is the expectation of this value. The values are generated for period January 1990 - December 2013 from the variables estimated by the datasets obtained from Thomson Reuters Datastream, the US Bureau of Labor Statistics and the US Federal Reserve Bank.
25
Name C UPR UTS UI UMP F-statistic DW Adj. 𝑹𝟐
T. Rowe Price -0.0189 -5.4927** 0.1522 -0.2767 -1.5960 4.9339** 1.9881 0.0522 (0.9411) (3.4786) (0.1958) (0.0893) (1.0698)
Table 6: The coefficients of the premia associated with each of the macroeconomic variables, as well as some descriptive statistics of the model; the F-statistic, Durbin-Watson-statistic and the adjusted r-squared. C represents the constant in the equation 𝑅𝑖,𝑡− 𝑅𝑓,𝑡= 𝛼 + 𝛽𝑖,𝑈𝑃𝑅∗ 𝑈𝑃𝑅𝑡+ 𝛽𝑖,𝑈𝑇𝑆∗ 𝑈𝑇𝑆𝑡+ 𝛽𝑖,𝑈𝐼∗ 𝑈𝐼𝑡+ 𝛽𝑖,𝑈𝑀𝑃∗ 𝑈𝑀𝑃𝑡, UPR the variable unexpected change in the risk premium, UTS the variable unexpected change in the term structure, UI the variable unanticipated change in rate of inflation, UMP the unanticipated growth in industrial production. The second and the fourth row in the table depict the coefficients in the equation; i.e. 𝛽𝑖,𝑘, whereas the bracketed values found in the third and fifth column represent the t-statistics associated with the coefficients. * implies that the coefficient or model is significant at a 5% threshold, whereas ** indicates that it is significant at a 1% level. The regressions are run on the variables calculated from the dataset retrieved from Thomson Reuters Datastream, US Bureau of Labor Statistics and the US Federal Reserve Bank for the period January 1990 - December 2013.
Name Period C UPR UTS UI UMP F-statistic DW Adj. 𝑹𝟐
26 Table 8: The coefficients of the premia associated with each of the macroeconomic variables, as well as some descriptive statistics of the model; the F-statistic, Durbin-Watson-statistic and the adjusted r-squared. C represents the constant in the equation 𝑅𝑖,𝑡− 𝑅𝑓,𝑡= 𝛼 + 𝛽𝑖,𝑈𝑃𝑅∗ 𝑈𝑃𝑅𝑡+ 𝛽𝑖,𝑈𝑇𝑆∗ 𝑈𝑇𝑆𝑡+ 𝛽𝑖,𝑈𝐼∗ 𝑈𝐼𝑡+ 𝛽𝑖,𝑈𝑀𝑃∗ 𝑈𝑀𝑃𝑡, UPR the variable unexpected change in the risk premium, UTS the variable unexpected change in the term structure, UI the variable unanticipated change in rate of inflation, UMP the unanticipated growth in industrial production. The values in columns 2-6 depict the coefficients in the equation; i.e. 𝛽𝑖,𝑘. * implies that the coefficient or model is significant at a 5% threshold, whereas ** indicates that it is significant at a 1% level. The regressions are run on the variables calculated from the dataset retrieved from Thomson Reuters Datastream, US Bureau of Labor Statistics and the US Federal Reserve Bank. The first period, displayed in panel a, consists of the months January 1995 - April 2001, where the data from January 1980 - December 1994 is used for estimation of IP and I by the ARIMA models. The second period, displayed in panel b, consists of the months May 2001 - September 2007, where the data from January 1980 - April 2007 is used for estimation of IP and I by the ARIMA models. The third period, displayed in panel c, consists of the months October 2007 - December 2013, where the data from January 1980 - December 2001 is used for estimation of IP and I by the ARIMA models. Each of the ARIMA models for IP and I in each of the periods is based on its individual Schwarz Information Criterion. The values displayed between brackets are the ranks of the coefficients with respect to the total sample of sectors.
Panel a. Period 1.
Name C UPR UTS UI UMP F-statistic DW Adj. 𝑹𝟐
Energy -0.0027 -6.9195** -0.8917 4.7556 -0.1287 4.1423** 1.9614 0.1452 {6} {1} {2} {4} {5} Financials -0.0881 -0.4455 5.0808 6.6161 -1.5183 2.4377 1.9463 0.0721 {1} {5} {6} {5} {3} Industrials -0.0184 -3.8403 -0.2484 7.1368 -0.0167 0.8917 1.9577 -0.0059 {3} {3} {4} {6} {6}
Oil and Gas -0.0169 -5.2735* -0.8457 -0.4262 -1.6440 1.8601 2.1256 0.0444
{5} {2} {3} {3} {2}
Retail food -0.0322 0.1487 -0.0283 -0.9705 -2.4363 0.5392 1.9858 -0.0255
{2} {6} {5} {2} {1}
Utilities -0.0180 -1.4687 -1.2845 -3.4003 -0.4072 0.2318 1.9869 -0.0433
27 Panel b. Period 2.
Name C UPR UTS UI UMP F-statistic DW Adj. 𝑹𝟐
Energy -0.0113 -4.1620 -1.1517 -1.8762 -0.4654 0.6412 2.0770 -0.0195 {5} {5} {3} {5} {3} Financials -0.0030 -7.9216 -1.9442 -4.4989 -0.6399 1.5911 2.2010 0.0306 {6} {1} {1} {3} {2} Industrials -0.0277 -5.2435* -0.3909 -7.4595** -0.3576 7.1646** 2.1489 0.2474 {1} {4} {5} {1} {4}
Oil and Gas -0.0240 -3.0474 -0.1060 -2.4625 0.4232 2.5357* 2.3272 0.0757
{2} {6} {6} {4} {5} Retail food -0.0185 -6.2061 -0.7288 -4.9955 0.1634 2.2183 2.4245 0.0610 {4} {3} {4} {2} {6} Utilities -0.0232 -6.2670 -1.3457 -1.3244 -1.3347 0.9684 1.7825 -0.0017 {3} {2} {2} {6} {1} Panel c. Period 3.
Name C UPR UTS UI UMP F-statistic DW Adj. 𝑹𝟐
Energy -0.0281 -2.7610** 0.6097 -3.6900* 0.1618 4.3534** 2.0583 0.1552 {5} {6} {2} {2} {6} Financials -0.0514 -3.4811** 1.3791 -0.0405 -1.2777 4.3392** 1.9237 0.1547 {2} {5} {5} {5} {2} Industrials -0.0536 -3.4877** 1.4143 -3.8613 -1.2039 3.9463** 2.1086 0.1390 {1} {4} {6} {1} {3}
Oil and Gas -0.0343 -4.5369** 0.7320 -0.4299 -0.5248 5.0961** 2.0492 0.1833
{4} {3} {3} {4} {5}
Retail food -0.0294 -10.7502** -0.3379 23.1872** -4.5060 6.3818** 2.7188 0.2277
{6} {1} {1} {6} {1}
Utilities -0.0423 -5.3253** 0.9417 -0.5358 -0.5859 4.1290** 2.0019 0.1464
