The Determinants of the Market Risk Premium
Master Thesis Finance
Second Semester 2014-2015
June 25, 2015
Abstract:
This paper addresses the factors that influence the market risk premium. This study examines monthly data of stock market indices in six European OECD countries in the period 1995 – 2014. The results of the simple and multiple regression analysis indicate that the risk-free rate, market volatility, dividend yield and price to earnings ratios are the primary determinants for explaining the market risk premium. In addition, the sample is split up into two sub samples to compare the pre-crisis and post-pre-crisis effect on the market risk premium. The difference is not significant. Furthermore, a static model is conducted to test the ability to predict the market risk premium. I find weak evidence to forecast the market risk premium on a short and long time horizon. The former is consistent with previous empirical literature, while the latter contradicts literature. An explanation of this contradiction is the relatively short time period of the data and the usage of different geographical markets.
Geert de Weerd
S2587165
2 1. Introduction
The collapse of the global financial crisis resulted in uncommon and turbulent markets and led stock investors to safer alternatives (Blake et al., 2012). This unusual behavior of the stock market increases the volatility of the market risk premium (MRP), which is defined by Fernandez (2004) as the incremental return required by investors over the risk-free rate to invest in the stock market (Duan and Zhang, 2014). Harris and Marston (2013) suggest that the uncertainty of investors triggers the MRP to shift upwards. In contrast, Brealey et al. (2011) expect a decrease in the MRP as the economy is recovering and more stable.
This study provides key insight into the determinants of the MRP. The MRP is one of the most important topics in valuation and portfolio theory, as investors estimate the cost of equity and determine the optimal allocation of assets with the key valuation parameter. An erroneous estimation of the market premium has a substantial impact on the cost of equity and any investment (Damodaran, 2014). The dispersion in the MRP can be explained by the variation in the explanatory variables or the cyclical behavior of the MRP (Harris and Marston, 2013). Furthermore, Harris and Marston (2013) assert that a constant MRP does not fully capture important changes in return requirements. Bätje and Menkhoff (2013) also emphasis that the fluctuating MRP is related to business peaks and troughs. The time-varying MRP is consistent with the theoretical implication of asset pricing models (e.g. CAPM), as perceptions of expected return can change.
Several authors assert that the risk-free rate, the dividend yield and the price to earnings ratio significantly influences the magnitude of the MRP (Dimson et al, 2008). However, the remaining multitude of factors are still part of a controversial debate (Bätje and Menkhoff, 2013). These additional factors are discussed in this paper. Moreover, Harris and Marston (2013) emphasizes that the MRP is conditional on a number of macroeconomic features in the economy. However, their research did not provide the answer to the key question to academics:
What are the determinants of the market risk premium for major European countries in the period 1995 – 2014?
In addition, the research question is answered with the following sub-questions:
What are the differences and the relations between the market risk premium and the determinants before and after the start of the financial crisis?
What are the differences and the relations between the market risk premium and the determinants in bull and bear markets?
3 magnitude of the MRP in bull and bear markets for the U.S. However, no recent study differentiates between states of the business cycle in major European countries and relate it to the magnitude of the MRP. This paper investigates whether there is a time-varying MRP and what the sign and magnitude this premium is across business peaks and troughs. While most research focuses on U.S. data, this paper extends previous literature by using recent European data after the global financial crisis. In addition, the 3-month German treasury bills declined after the global crisis to record low interest rate of -0.379 on 10 May 2015. Previous empirical literature find an inverse relation between the risk-free rate and MRP. The exceptional low risk-free rate and uncertainty of investors in the aftermath of the global financial crisis induces many academics and financial professionals to review the MRP.
In this paper, the actual MRP is computed with the historical approach, which is the most applied approach and consistent with the method of numerous authors (Mehra and Prescott, 1985; Dimson et al., 2008). A simple and multiple regression analysis is conducted to test for a contemporaneous relation between the MRP and the explanatory variables. Furthermore, a static model is conducted to test the ability to predict the MRP in the next month or year, since several authors find predictive explanatory power. The primary reason for the use of the time lags is to distinguish between the historical MRP and the expected future MRP (Arnott and Bernstein, 2002). In addition, financial professionals want to know what expected MRP they need to use for the following period.
The results show that the historical MRP is predominantly determined with the risk-free rate, the historical volatility of the stock market, the average dividend yield and the average price/earnings ratio. Furthermore, the magnitude of the MRP is significantly higher in bull markets. However, this paper finds no evidence of a different MRP in the pre-crisis and post-crisis period. I also find weak evidence to predict the MRP on a short and long time horizon. The former is consistent with prior empirical research, while the latter contradicts research. Only the average dividend yield of the stock market and the risk-free rate are able to explain the MRP on a short time horizon, which is not coherent with the efficient market hypothesis. The results provide also guidance to financial professionals. Professional can predict the magnitude of the MRP in the next month with the average dividend yield and the risk-free rate.
4 2. Literature
2.1 Theoretical literature
The market risk premium is important for multiple asset pricing models. In the CAPM, the MRP is calculated as the expected return on the market subtracted by the risk-free rate. However, the asset pricing models implicitly assumes that the MRP is known, while it remains extremely challenging for both academics and financial professionals to estimate the MRP. In addition, the exact magnitude of the MRP is not available to anyone as the elusive parameter is non-observable.
The risk-free rate directly relates to the estimation of the MRP. Damodaran (2013) emphasizes that an investment is risk-free if there is (1) no variance around the expected return and (2) no default risk. A theoretical explanation for the relation between the risk-free rate and the MRP is provided by McKenzie and Partington (2013). The authors posit that if the MRP is equal to the expected return minus the risk-free rate, than per definition formula (1) also holds:
𝐸(𝑅𝑚) = 𝑀𝑅𝑃 + 𝑅𝐹 (1)
where, 𝐸(𝑅𝑚) is the expected return on the market, 𝑀𝑅𝑃 is the Market risk premium and 𝑅𝐹 is the
risk-free rate, respectively. There exist an inverse relation between the MRP and the risk-free rate if the expected returns are assumed to be constant1.
The theoretical relation between the MRP and the volatility of the stock market can be derived from the paper of Merton (1980). Merton presents the intertemporal CAPM (ICAPM) and estimates the MRP with a theoretical relation between expected returns and the conditional variance. This positive relation between both variables is also coherent with financial theory, which assert that risk averse investors are rewarded for taking additional risk.
The theoretical relation between the MRP and average dividend yield is ambiguous. The Dividend Irrelevance theory of Modigliani and Miller (1961) posits that investors are indifferent between payments in dividend or capital gains under the assumption of frictionless and perfect capital markets. However, the Gorden/Lintner theory or synonymously ’the bird in the hand theory’ asserts that the dividend yield is a crucial measure for the returns of investors. The semi-strong form of the Efficient Market Hypothesis (EMH) asserts that markets are efficient and fully reflect all publicly available information (Hillier et al., 2011). The theory implies that neither technical analysis, which is the prediction of future prices with historical data, nor fundamental analysis or synonymously the analysis of financial information of corporations, would generate returns greater
1 More theoretical explanations for the negative relation between the MRP and the risk-free rate are provided
5 than randomly selected portfolios with identical risk (Malkiel, 2003). Dividend yields and other valuation ratios should not be able to predict movements in the stock market (Campbell and Shiller, 1998).
The magnitude of the MRP has also huge impacts on the optimal allocation of assets. Portfolio theory asserts that the allocation to stocks positively relates with an increase in the MRP.2 Investors allocate more assets to stocks when the risk premium is higher or as there is a steeper risk-reward curve. The steep curve indicates more incremental return for each unit of risk.
2.2 Empirical literature
2.2.1 Magnitude of the MRP
The historical market risk premium is the additional return of the stock market over the riskless securities and is commonly used as a guesstimate of the expected market premium (Fernandez, 2004). In addition, Mayfield (2004) demonstrates that estimating the MRP with the historical excess returns of the stock market is the most frequently employed method in practice. Ibbotson and Chen (2003) also emphasizes that the historical approach is a good indicator of the MRP and that on average the market is right. Several authors adopt this method for determining the magnitude of the MRP on an annual basis (Mehra and Prescott, 1985; Dimson et al., 2008). Mehra and Prescott (1985) computed historical returns from data in the period 1889 – 1978 and find a MRP of 6.18% for U.S. companies, while Ibbotson and Chen (2003) estimates a slightly lower MRP of 5.90% in 1926 - 2000. Damodaran (2013) also estimates the arithmetic average of the market premium for several time periods, and find a decreasing trend of 7.55%, 5.38% and 3.12% for the period 1928 - 2011, 1962- 2011 and 2002 - 2011 respectively. To exacerbate the issue of the dispersion even further, Fernandez (2009) reviewed 100 financial textbooks and stipulates a variation in the MRP of 3 – 10 percentage-point. An explanation of the dispersion is (1) the cyclical behavior of the MRP, (2) the discrepancy in calculating the MRP, (3) the dissimilarities or variation in explanatory variables and (4) the different methods for determining the magnitude of the MRP (Fernandez, 2013). A wrong or haphazard estimation of the market premium has a substantial impact on the cost of equity and any investment (Damodaran, 2013). The Absolute Strategy (ASR), Europe’s leading
2
The weight in stocks is determined with the following formula (Elton et al, 2014):
𝑊 = 𝑇
2
𝐸(𝑟) − 𝑅𝐹
𝜎2
6 macro strategy provider, estimates the MRP in several European countries, which data is available on Datastream. The average MRP of six OECD countries and the MRP of the Eurozone (measured by Eurofirst100) for the period of interest (1995 – 2014) is respectively, 3.9% and 3.8%. These findings of the ASR indicate that the MRP in major European countries are almost equal to the MRP of Europe.
2.2.2 Determinants of the MRP
The dispersion in the MRP can be caused by the variation in the explanatory variables or the cyclical behavior of the MRP. One implicit variable in the MRP is the risk-free rate. Harris and Marston (2013) emphasizes that interest rates below the historical average causes the MRP to shift upward. For data in the second quarter of 2011, the authors find an upward shift of the MRP with 1.37% for a simultaneous downward change of the interest rate with 2.06%. Although there are a host of caveats, like the short time period, they certainly indicate a strong negative relation between the MRP and the risk-free rate. Ang and Beakert (2007) also find an inverse relation between both variables and that the risk-free rate is able to predict the magnitude of the MRP on a short horizon.
One determinant of the MRP is the volatility of the stock market. Several authors use the historical volatility as a proxy for the risk of the market index (Pesaran and Timmermann, 1995; Dimson et al., 2011). Sanvicente and Carvalho (2012) use monthly data on the S&P 500 and asserts that the MRP is influenced by the level of the volatility in 1995 - 2010. The relationship between the MRP and the volatility is empirically ambiguous. Baillie and Degennaro (1990) find a positive relation between both variables, while Glosten et al. (1993) and Campbell (1987) show a negative relation. Bollerslev et al. (2009) emphasizes that the empirical literature provides no robust positive relation and is inconclusive about the sign of the relationship. The mixed results can be explained with the findings of Girard et al. (2001) who investigates the risk-returns relation before and after the Asian crisis of 1997. The authors find a positive relation in an upward market trend and a negative relation in a downward market trend.
7 (2006) find that ability to predict the MRP with the dividend has weakened, while Ang and Bekaert (2007) find no statistical significance. The second financial parameter is the average price to earnings (P/E) ratio of the stock market. The P/E ratio is an important variable for the MRP, since a high (low) P/E ratio implies a high (low) growth of a company in the future. Several authors find that the P/E ratio of the market index has forecasting power of future stock returns (Campbell and Shiller, 1998; Grinold et al., 2011). Hirshleifer et al. (2008) explains that high 𝑃/𝐸 ratios are associated with the overvaluation of the aggregated stock market. Damodaran (2012) confirms this explanation and find a negative relation between the P/E ratios and the MRP.
McQueen and Roley (1993) find that the stock market respond differently on macroeconomic factors dependent on the state of the economy.More specifically, the MRP increases (decreases) in bull (bear) markets for higher than expected economic factors. The first macro-economic factor which influences the MRP is the growth in GDP. Cooper and Priestley (2009) emphasizes that GDP growth captures important variations of the business cycles. Faugère and Erlach (2006) investigates the MRP with the S&P 500 in 1926 - 2001 and find that the MRP is a function of GDP growth. Ritter (2004) find a negative relation between the MRP and the GDP growth for 16 developed countries over the period 1900 - 2002. The implication of this finding is that GDP growth exacerbates the returns on the stock market and that the MRP is a counter-cyclical variables. Siegel (1998) and Shackman (2006) find similar results and explains that largest corporations quoted on market indices are multinationals, who operate worldwide rather than domestically3.
Damodaran (2014) asserts that governments in European countries were able to maintain a relatively stable risk-free rate before the global financial crisis. However, the global financial crisis changed the responses of government spending in European Countries as interest rates sharply increased to almost 30 percentage-point in Greece and declined in other countries. Pástor and Veronesi (2012) demonstrates that perceptions about the government policy have both economic and non-economic effects. The uncertainty of the economic effects increases both the volatility of the market and the MRP. Thorbecke (1997) find that investors require higher returns for government interventions and that the monetary policy is a priced risk factor. The money supply of the government, measured by the M1, which excludes government reserves, is a macro-economic factor which capture these impacts of monetary policy. Wong et al. (2005) examines the effects of the M1 on the market returns in Singapore and find a positive relationship before the Asian crisis. However, the relation between both variables disappeared after the financial crisis. The authors find similar
3
8 results for the crisis of 1987 in the United States. McQueen and Roley (1993) also emphasizes that the relation between the M1 and the MRP is conditional on the state of the economy.
Several studies find a positive relation between the inflation and the risk premium (Brandt and Wang, 2003; Connolly et al, 2015). Brandt and Wang (2003) provides evidence of an increase in the MRP if the unexpected inflation growth is higher than anticipated. McQueen and Roley (1993) use the Consumer Price Index (CPI) as a proxy for inflation and find a positive relation between the stock returns and the inflation in optimism (bull) markets. However, the authors find no significant relation in pessimistic markets. They conclude a time-varying relation that is conditional on the state of the economy.
Finally, Modugno (2012) asserts that investors expect the highest market returns at the top of the bull markets and the lowest returns at the bottom of bear markets. Hedi and Fredj (2010) investigates several periods of the crisis in the US and find that the crisis have significant impact on the MRP. A decrease of the MRP after the start of the global crisis is expected, as the returns of the European market indices are lower than before the crisis (Damodaran, 2013).
2. Data and Research Methods 2.1 Formulation of Hypotheses
With this paper, I will determine the factors that influence the magnitude of the MRP. The literature posits that the primary determinants of the MRP are the average financial ratios which serve as a proxy for business conditions and several macro-economic variables. Based on the literature, I have the following testable hypotheses:
𝑯𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔 𝟏: There is a negative relation between the MRP and the risk-free rate. 𝑯𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔 𝟐: There is a relation between the MRP and historical volatility.
𝑯𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔 𝟑: There is a positive relation between the MRP and the average dividend yield. 𝑯𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔 𝟒: There is a negative relation between the MRP and the average price/earnings ratio. 𝑯𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔 𝟓: There is a negative between the MRP and the Gross Domestic Product.
𝑯𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔 𝟔: There is a relation between the MRP and the money supply (M1).
𝑯𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔 𝟕: There is a positive relation between the MRP and the consumer price index.
A couple of additional hypotheses are addressed to assess whether the magnitude of the MRP and the relation with the determinants is affected by the crisis and market peaks/troughs:
9 𝑯𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔 𝟗: The relation between the several determinants and the MRP is different before and
after the financial crisis.
𝑯𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔 𝟏𝟎: The magnitude of the MRP is higher in bull markets.
𝑯𝒚𝒑𝒐𝒕𝒉𝒆𝒔𝒊𝒔 𝟏𝟏: The relation between the several determinants and the MRP is different when
differentiating between bull and bear markets.
2.2 Methodology
The historical returns provides an indication of the future market as the market is influenced by the historical performance (Pratt and Grabowski, 2014). In addition, taking the average of historical excess returns of the stock market remains the most commonly used method for estimating the MRP (Mayfield, 2004).The ex-post instead of ex-ante approach is applied, which means that realized historical returns are calculated instead of expected return. The arithmetic average method is applied in this research, since it is the most widely used approach and provides the best estimate of the future MRP (Pratt and Grabowski, 2014; Koller et al. 2010). The MRP is calculated by subtracting the risk-free rate from the market return on a monthly basis, which is consistent with several authors (Campbell and Thompson, 2008; Goyal and Welch, 2008) The formula is as follows:
𝑀𝑅𝑃𝑡 = 𝑅𝑖,𝑡− 𝑅𝐹𝑡 (2)
where, 𝑀𝑅𝑃𝑡 is the market risk premium at month t, 𝑅𝑀,𝑗,𝑡 is the monthly market returns on index i
at month t, and 𝑅𝐹𝑡 is the risk-free rate at month t, respectively. The data is collected using the total return index, which includes the reinvestments of the dividends in the indices, otherwise the inferences are affected with a downward bias of the returns. The monthly returns on the stock market are calculated with reference to the market index of that country and calculated with formula (3):
𝑅𝑖,𝑡 = 𝐿𝑛 𝑃𝑖,𝑡
𝑃𝑖,𝑡−1 (3)
where, 𝑅𝑖,𝑡 is the monthly market returns on index i at the end of month t and 𝑃𝑖,𝑡 is the total
return index of market index i at the end of month t. The MRP is calculated with country specific premiums. Only the data of major European countries are retrieved from Thomson Reuters Datastream, as they represent 86% of total market capitalization in Europe (ASR, 2014)4. The stock
4 The major European country Italy is excluded in this paper due to data availability of the explanatory
10 market indices are the FTSE100 index, DAX30, IBEX35, AEX, CAC40, and SMI, respectively5. All the Datastream codes are presented in Appendix A.
The risk-free rate is extremely important for the magnitude of the MRP. Mikhel (2012) emphasizes that none of the European governments control the money supply. Therefore, there is some default risk in all European bonds. The government securities with the minimum risk are German Treasury bills and bonds. Harris et al. (2012) find that German bills trade at large volume and are highly liquid. Moreover, Germany is AAA credited according to the Sovereign rating list and consist of the least credit risk (S&P, 2015). Several authors use the yield on a 10-year treasury bond as a proxy for the risk-free rate for the valuation of long term investments (Grinold and Kroner, 2002; Damodaran, 2014). However, the 10-year T-bond is not consistent with the duration of the monthly data. The one-month T- bill is only available in Datastream In the 21th century. However, the MRP needs to be calculated over a long period of time, as it reduces the estimation error (Koller et al., 2010; Damodaran, 2014). In contrast, a too long period indicates that the MRP is influenced by the survival bias and that markets do not change (Brown et al., 1995).The closest match to the duration of the monthly data is the 3-month German treasury bills. The starting point of these bills (1995) are the beginning of the sample. Dimson et al. (2008) and Li and Liu (2010) uses the yield on the U.S. treasury bills for developed countries in Europe as a proxy for the free rate and not the local risk-free rates. Consistent with the research of Damodaran (2008), this paper uses the yield on the German treasury bill as a proxy for the risk-free rate, since there is less default risk than other European bills. The 3-month German treasury bill is retrieved from Datastream. The yields are annually compounded and converted into a monthly basis with formula (4):
𝑅𝐹𝑡=𝐿𝑛 (1 + 𝑅𝐹3𝑚
100 )
12 (4)
where, 𝑅𝐹3𝑚 is the annually compounded 3-month German yield, and 𝑅𝐹𝑡 is the continuously compounded risk-free rate on a monthly basis6.
5
FTSE 100 represents the market index of the United Kingdom and is the most widely used UK market indicator, the DAX30 represents the market index of Germany, IBEX35 represents the market index of Spain, the AEX index is the market index of the Netherlands, CAC40 represents the market index of France, and SMI represents the market index of Switzerland, respectively. These indices are equivalent to the market indices in the paper of Lu and Liu (2010).
11 According to previous empirical literature, a low risk-free rate should increase the MRP. A simple regression model is conducted to test the impact of the simultaneous changes in the interest rate on the MRP.7 The formula for the simple regression model is as follows:
𝑀𝑅𝑃𝑡= 𝛼 + 𝛽𝑅𝐹𝑡+ u𝑡 (5)
where 𝛼, 𝛽 and 𝑢𝑡 are the intercept, regression coefficient, and the random disturbance term at
month t, respectively. Bekaert and Ang (2007) find significant evidence for the prediction of the MRP with the risk-free rate at a short horizon. Formula (6) is used to test this explanatory power of the risk-free rate on a short horizon:
𝑀𝑅𝑃𝑡 = 𝛼 + 𝛽𝑅𝐹𝑡−1+ u𝑡 (6)
where, 𝛽𝑅𝐹𝑡−1 is the lagged risk-free rate. Both simple regression models are conducted to test hypothesis 1, or the relation between the risk-free rate and MRP. The sample is divided into two sub samples, to compare the pre-crisis and post-crisis effects on the MRP, which started in August 20078.
The starting point of the crisis is exactly the same as in the paper of Haverstad (2010).
Consistent with the findings of numerous authors, this paper uses the absolute level of the average dividend yield and the average price/earnings ratio of the stock market (Abel, 1988; Ang and Beakert, 2007). The ratios have a tendency to return (mean revert) to the historical average. The Augmented Dickey-Fuller (ADF) test is conducted to test for a stationary process of the other explanatory variables. The Kwiatkowski-Phillips-Schmidt-Shin (KPSS) test is also conducted to complement the ADF unit root test. The stock return volatility is measured on a monthly basis. I use the daily returns of the market indices to estimate the monthly volatility of the stock market. A more precise estimate of the volatility in month t is obtained by using the daily returns, because the volatility of the stock market is not constant (French, Schwert and Stambaugh, 1987). First, all the daily returns of the market indices are retrieved from Datastream and converted into a log level. The standard deviation of this return is calculated for all the trading days in month t, which is then called the daily historical volatility. The volatility of the daily stock returns of month t is multiplied with the square root of the number of trading days in month t9. The formula is provided in the paper of Ederington and Guan (2004).
7
Although I find some outliers in the data, they are not extreme, since it considers the stock market index and not specific companies.
8
Starting from August 2007, there are large negative returns for all market indices. This market downfall indicates the start of the global crisis.
9 All the trading days are counted with Microsoft Excel. The number of trading days (5218) are divided by the
12 The average monthly dividend yield (DY) is retrieved from Datastream and is based on annual dividend rates and excludes once-off or synonymously special dividends10. The 𝐷𝑌 is expressed as the dividend per share as a percentage of the share price (Datastream, 2008). The 𝐷𝑌 of the index is the average of the individual dividend yields weighted by market value (Datastream, 2008). The formula in footnote 10 indicates that the 𝐷𝑌 changes monthly, even though companies only pay dividends quarterly/yearly, because the stock prices changes. The Price/Earnings ratio is calculated as the constituents share price divided by the per-share earnings11. The P/E of the index is the average of the individual P/E ratios of the constituents weighted by market value (Datastream, 2008).
The GDP is available in Datastream on a quarterly basis for the countries where the market indices are located. The GDP is converted into monthly frequency using the ‘’quadratic-match’’ function by the econometric software EViews, which performs a local quadratic interpolation between the known quarterly data.12 The method is consistent with Cheng et al. (2012), who also uses this function to interpolate quarterly data into monthly data. The natural logarithm is taken to measure the growth in GDP. The money supply of the government, measured by the M1 is also retrieved from Datastream. The data on CPI are converted into a log level, seasonally adjusted and used as a proxy for inflation. This method is consistent with McQueen and Roley (1993), who also uses seasonally adjusted log changes in the CPI.
The risk-aversion of the investors is also an important variable to consider. However, Damodaran (2014) emphasizes that there exist a significant anxiety among economic researchers to relate risk aversion to the MRP. Determining the MRP with the risk aversion of the investors does not explain the MRP accurately. In addition, in neo-classical models, the risk premium are often set equal to the product of relative risk aversion. The risk-aversion is therefore not included in the model.
10
The formula is as follows;
𝐷𝑌𝑡=
∑ (𝐷𝑌𝑛1 𝑗𝑡∗ 𝑀𝑉𝑗𝑡)
∑ 𝑀𝑉𝑛1 𝑗𝑡
where, 𝐷𝑌𝑡 is the dividend yield of the index at month t, 𝐷𝑌𝑗𝑡 is the dividend yield of stock j at month t, 𝑀𝑉𝑗𝑡 is
the market value of stock j at month t, and 𝑛 is the number of constituents, respectively. 11
The formula is as follows;
𝑃𝐸𝑡=
∑ 𝑀𝑉𝑛1 𝑗𝑡
∑ (𝑀𝑉𝑛1 𝑗𝑡⁄𝑃𝐸𝑗𝑡)
where, 𝑃𝐸𝑡 is the Price/Earnings ratio of the index at month t, and 𝑃𝐸𝑗𝑡 is the Price/Earnings ratio of stock j at
month t, respectively.
12 The quadratic polynomial is performed by taking a point of the original data (quarterly) and convert it to
13 A multiple regression analysis is conducted in order to test for the collaborative impact of the variables on the MRP. The Efficient Market Hypothesis asserts that lagged valuation ratios are unable to predict the change in the MRP in the next month. Therefore, a contemporaneous relation between the explanatory variables and the MRP is examined. A change in the independent variables causes an instant change in the dependent variables. So the first multiple regression analysis focuses on simultaneous changes in the MRP and explanatory variables. The formula for the multiple regression analysis including no time lags is as follows13 :
𝑀𝑅𝑃𝑡 = 𝛼0+ 𝛽1𝑉𝑂𝐿𝑡+ 𝛽2𝐷𝑌𝑡 + 𝛽3𝑃/𝐸𝑡+ 𝛽4 𝐺𝐷𝑃𝑡+ 𝛽5𝑀1𝑡 + 𝛽6𝐶𝑃𝐼𝑡+ ut (7)
where, 𝑉𝑂𝐿 is measured as the historical volatility of the specific market index in month t calculated on daily returns, 𝐷𝑌 is the average dividend yield of the market index, 𝑃/𝐸 is the average price/earnings ratio of the market index, 𝐺𝐷𝑃 is the growth in the Gross Domestic Product of the country where the market index is located, 𝑀1 is the growth in the money supply or the monetary items available in the economy, and 𝐶𝑃𝐼 is the growth in the Consumer Price Index of the specific country, respectively.
A static regression model uses lagged explanatory variables to model the dependent variable. Several authors find evidence of the ability to predict the risk premium with the dividend yield and Price/Earnings ratios by using time lags (Campbell and Diebold, 2009; Bätje and Menkhoff, 2013; Neely et al., 2014). The primary reason for the use of the time lags is to distinguish between the actual MRP and the expected future MRP (Arnott and Bernstein, 2002). In addition, financial professionals want to know what expected MRP they need to use for the following period. Goyal and Welch (2003) asserts that forecasting the risk premium with the historical risk premium is the best method. The multiple regression model including time lags is as follows:14
𝑀𝑅𝑃𝑡 = 𝛼0+ 𝛽1𝑉𝑂𝐿𝑡+ 𝛽2𝐷𝑌𝑡−1 + 𝛽3𝑃/𝐸𝑡−1+ 𝛽4𝐺𝐷𝑃𝑡+ 𝛽5𝑀1𝑡 + 𝛽6𝐶𝑃𝐼𝑡 + ut (8)
where, 𝐷𝑌𝑡−1 is the lagged average dividend yield of the stock market, and 𝑃/𝐸𝑡−1 is the lagged
average price to earnings ratio.
A one-tailed independent sample t - test is conducted to provide statistical evidence about hypothesis 8 and 10. The t-statistic is determined according to the following formula (Keller, 2009):
13
I also test for the term spread, which is defined as the difference between the yield on the 10-year T-bond and the yield on 3-month T-bills. This approach is consistent with the measurement of Goyal and Welch (2008) and Faugère and Erlach (2006). However, I find no significant coefficients. Therefore, the term spread is not included in the regression model.
14
14 𝑡 = 𝐴𝑀𝑅𝑃2𝑡− 𝐴𝑀𝑅𝑃1𝑡 √𝜎 22 𝑁2+ √𝜎 1 2 𝑁1 (9)
where, 𝐴𝑀𝑅𝑃2𝑡 and 𝐴𝑀𝑅𝑃1𝑡 is, respectively, the average market risk premium per country before
and after the start of the financial crisis, 𝑁1 and 𝑁2 are the number of months in the period, and 𝜎 12
and 𝜎 22 are the variances of the average market risk premium for sub sample 1 and sub sample 2. The
variable σ is calculated as follows; 𝜎2 = 1
𝑁2 ∑ 𝜎2𝑀𝑅𝑃𝑗𝑡 (10)
where, 𝜎2 is the variance of the MRP.
According to Campbell and Diebold (2009) the MRP is time varying. However, it is intriguing to find out if the MRP is a pro-cyclical or counter-cyclical variable, because the MRP can then be related and forecasted with regards to the growth or state of the economy. The peaks and troughs are determined by the Economic Cycle Research Institute (ECRI). The ECRI publishes the top of the bull market and the bottom of the bear market of major European countries each year and are presented in Appendix B15. The ECRI identify several characteristics of the market in order to distinguish between the period of market optimism (bull market) and the period of market pessimism (bear market). The leading indicators are (1) the state of economy, measured by macro-economic growth in the country and (2) several aspects of major sectors.
2.3 Descriptive Statistics
Many authors present the descriptive statistics of the MRP measured by the difference between the market return and both T-bonds and T-bills (McQueen and Roley, 1993; Dimson et al., 2011). Both measures are presented in this paper to compare the results. The descriptive statistics of the market return, market return – T. bills, and market returns - T. bonds for the full sample are presented on a monthly basis in table 1, table 2 and table 3, respectively. The descriptive statistics of the pre-crisis and post-crisis data are presented in Appendix C. The mean MRP is positive, which implies that investors in European stock indices receive a compensation for their exposure to the stock market risk. The MRP is higher (lower) when bills (bonds) are used as a proxy for the risk-free rate, which is consistent with the earlier finding of Dimson et al. (2011) and the liquidity preference theory. The mean MRP is the highest (lowest) in the pre-crisis (post-crisis). The aggregated stock market returns are negative skewed, which is consistent with the findings of several authors (Albuquerque, 2010). The first sub
15 In addition, the duration of a complete business cycle, business peak or trough, must be at least 15 months
15 Table 1
Descriptive statistics
This table represents the descriptive statistics of the monthly return of the stock market indices for the full sample (1995 – 2014). FTSE100 Index represents the market index of the United Kingdom and is the most widely used UK market indicator, DAX30 is the market index of Germany, IBEX35 represents the market index of Spain, the AEX index is the market index of the Netherlands, CAC40 is the market index of France, and SMI represents the market index of Switzerland, respectively.
FTSE DAX IBEX AEX CAC SMI
Mean 0.0062 0.0066 0.0080 0.0061 0.0059 0.0071 Median 0.0092 0.0161 0.0134 0.0148 0.0121 0.0127 Std. Dev. 0.0473 0.0706 0.0680 0.0682 0.0626 0.0566 Minimum -0.2193 -0.2847 -0.1962 -0.3336 -0.2508 -0.2842 Maximum 0.1569 0.2162 0.2479 0.2453 0.1926 0.2280 Skewness -0.8972 -0.9922 -0.2030 -1.2477 -0.8925 -0.8776 Kurtosis 6.8708 5.9824 3.9375 8.2587 5.7502 7.7747 Table 2 Descriptive statistics
This table represents the descriptive statistics of the MRP, where the 3-month treasury bills are used as a proxy for the risk-free rate for the full sample (1995 – 2014).
FTSE DAX IBEX AEX CAC SMI
16 Table 3
Descriptive statistics
This table represents the descriptive statistics of the MRP, where the 10-year treasury bonds are used as a proxy for risk-free rate for the full sample (1995 – 2014).
FTSE DAX IBEX AEX CAC SMI
Mean 0.0029 0.0034 0.0048 0.0028 0.0027 0.0039 Median 0.0065 0.0120 0.0104 0.0114 0.0090 0.0101 Std. Dev. 0.0473 0.0706 0.0680 0.0682 0.0626 0.0566 Minimum -0.2226 -0.2882 -0.1986 -0.3369 -0.2540 -0.2874 Maximum 0.1535 0.2129 0.2445 0.2420 0.1893 0.2247 Skewness -0.8917 -0.9917 -0.1995 -1.2488 -0.8908 -0.8800 Kurtosis 6.8383 5.9718 3.9341 8.2485 5.7230 7.7504
-80 -60 -40 -20 0 20 40 60 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 FTSE DAX IBEX AEX CAC SMI Figure 1
Magnitude of the MRP using historical returns
18 4. Results
4.1 Determinants of the market risk premium
The results from the simple bivariate regression model to examine the relation between the risk-free rate and the MRP are presented in table 4. The coefficients are corrected for the presence of heteroskedasticity with the white test. The negative coefficients are hardly significant when no time lags are applied. However, all coefficients are negative and significant at a 5% or 10% significance level with time lags (except for FTSE). The hypothesis 1 is rejected, which indicates a negative relation between the risk-free rate and the MRP. The results are consistent with Harris and Marston (2013), who asserts an inverse relation between the former and the latter. Harris and Marston (1992) emphasizes in an earlier study that a one percentage-point increase in the risk-free rate decrease the MRP approximately with a half percentage-point. However, the results in this paper indicate a stronger negative relation. For the Netherlands, the MRP decreases approximately with five percentage-point for a simultaneous increase in risk-free rate with one percentage-point. The impact is even higher in the following months. The results of Harris and Marston (1992) seems to be outdated for more recent data. One explanation could be the historical low interest rate, where an increase of the interest rate with one percentage-point is relatively extreme. The negative relation
Table 4
Relation between German risk-free rate and the Market risk premium
This table shows the results of the simple bivariate regression analysis to find the relation between the Market risk premium and the 3-month German treasury bills. The results of this table are based on formula (5) and (6). The standard deviations are corrected for heteroskedasticity with the white test. The minuses correspond to the amount of lags in months. The standard errors appear in parentheses, where ***,** and * correspond to 1%, 5%, and 10% significance level, respectively.
19 becomes stronger for the lagged variables. However, there are no significant coefficients using 12 time lags and they are therefore not presented in table 4. This result is consistent with the findings of Bekaert and Ang (2007), who find a predictability of the MRP with the risk-free rate only for horizons of less than a year.
The findings of both unit root tests are presented in Appendix E. The results indicate that the explanatory variables have a unit root and are non-stationary across indices16. The first differences
of the variables are taken to ensure a stationary process. Three correlation matrixes with the highest correlation between the explanatory variables are presented in Appendix F17. The potential multicollinearity problem is the relatively high negative correlation between the 𝐷𝑌 and the 𝑃/𝐸 ratio (-0.69 for the FTSE). However, statistical theory states that a problem occurs if the bivariate correlation exceeds the absolute value 0.8 (Bens et al., 2003). Moreover, I test for the Variance Inflation Factor (VIF) in EViews for potential multicollinearity problems between the explanatory variables. The values are below the threshold of ten, which indicates that we have no serious multicollinearity problems (O’Brien, 2007). The multiple regression is also conducted without the 𝐷𝑌 to test whether the coefficients and the significance levels change18.
The findings of the multiple regression analysis to examine the contemporaneous relation between the MRP and the explanatory variables for the full sample is presented in table 5. I test for heteroskedasticity and autocorrelation with the white test and Durbin Watson test, respectively, and find significant evidence of the presence of both. The coefficients are corrected for the presence of heteroskedasticity and autocorrelation with the Breusch-Godfrey test. 𝑉𝑂𝐿 and 𝑃/𝐸 are the primary determinants for explaining the magnitude of the MRP, since most of the variables are robust across indices. The 𝑉𝑂𝐿 of the market index FTSE specifies that if the market volatility goes down with one percentage-point, the MRP simultaneously increases with 0.8109, assuming the constancy of other explanatory variables. So an increasing MRP is associated with declining volatilities (Giot, 2005). There is abundant evidence to reject the hypothesis 2. The negative relation is not coherent with the theoretical Intertemporal CAPM model of Merton (1980). However, the result is consistent with the empirical finding of Glosten et al. (1993) and Campbell (1987). One explanation for the inverse contemporaneous relation between both variables is the volatility feedback effect. According to the volatility feedback effect, an unexpected increase in the volatility leads to an immediate decline in stock prices, since cash flows are discounted at a higher rate (Kanniainen and Piché, 2012).
16 Although the statistics from Switzerland indicate that the CPI is stationary, the statistics of other countries show overwhelming evidence that the variable is non-stationary.
17
The other correlation matrixes are available on request.
18 The 𝐷𝑌 is excluded in the multiple regression analysis to test the multicollinearity problem, because I find
20 Table 5
Determinants of the Market risk premium for the full sample
This table shows the results of the multiple regression analysis based on monthly data to find the determinants of the MRP in the period 1995 – 2014. The regression is as follows:
𝑀𝑅𝑃𝑡= 𝛼0+ 𝛽1𝑉𝑂𝐿𝑡+ 𝛽2𝐷𝑌𝑡 + 𝛽3𝑃/𝐸𝑡+ 𝛽4 𝐺𝐷𝑃𝑡+ 𝛽5𝑀1𝑡 + 𝛽6𝐶𝑃𝐼𝑡+ut
where, 𝑉𝑂𝐿 is measured as the historical volatility of the specific market index, 𝐷𝑌 is the average dividend yield of the market index, 𝑃/𝐸 is measured as the average price/earnings ratio of the market index, 𝐺𝐷𝑃 is the growth in Gross Domestic Product of the country where the market index is located, 𝑀1 is the growth in the money supply or the monetary items (excludes reserves) available in the economy, and 𝐶𝑃𝐼 is the growth in the Consumer Price Index of the specific country. The coefficients are corrected for the presence of heteroskedasticity and autocorrelation with the Breusch-Godfrey test. The standard errors appear in parentheses, where ***,** and * correspond to 1%, 5%, and 10% significance level, respectively.
The growth in GDP is negatively related with the MRP, which is consistent with the findings authors (Shackman, 2006; Siegel, 1998; Ritter, 2004. However, this finding is not robust across indices and inferences need to be interpreted with caution. This negative relation implies that the MRP is a counter-cyclical variable or relates negatively with the economy. Accelerating (decelerating) economic growth results in a lower (higher) MRP. There is a relation between the M1 and the MRP, but the results from table 5 do not indicate the sign of the relation with certainty. The mixed results can be explained by the findings of Wong et al. (2005), who find that the M1 is positively related to MRP before the Asian crisis, but not after the start of the crisis. The contemporaneous relation between 𝐶𝑃𝐼 and the MRP show no significant coefficients. Therefore, we are not able to reject hypothesis 7, which indicate that there is no relation between the 𝐶𝑃𝐼 and the MRP. The adjusted 𝑅2
FTSE DAX IBEX AEX CAC SMI
21 for the market indices of the full sample ranges from 0.1751 to 0.3703. The results of the multiple regression analysis without the 𝐷𝑌 is presented in Appendix G. The results show no major differences in the coefficients and significance levels. This finding indicate that there is no major multicollinearity problem.
The results of the model including time lags are presented in table 6. The results of table 6 indicate a strong positive relation between the 𝐷𝑌 and the MRP. These results are consistent with the findings of several authors (Bekaert et al., 2009). The null hypothesis regarding the relation between the dividend yield and the MRP can be rejected. The positive relation implies that the 𝐷𝑌 of the previous month has the ability to predict the MRP on a short horizon. The high average
Table 6
Determinants of the market risk premium
This table shows the results of the multiple regression analysis based on monthly data to find the determinants of the MRP in the period 1995 – 2014. The regression is as follows:
𝑀𝑅𝑃𝑡= 𝛼0+ 𝛽1𝑉𝑂𝐿𝑡+ 𝛽2𝐷𝑌𝑡−1 + 𝛽3𝑃/𝐸𝑡−1+ 𝛽4𝐺𝐷𝑃𝑡+ 𝛽5𝑀1𝑡 + 𝛽6𝐶𝑃𝐼𝑡 + ut
where, 𝑉𝑂𝐿 is measured as the historical volatility of the specific market index, 𝐷𝑌𝑡−1 is the mean
dividend yield of the market index with one lag, 𝑃/𝐸𝑡−1 is measured as the mean price/earnings ratio
with one lag, 𝐺𝐷𝑃 is the growth in Gross Domestic Product of the country where the market index is located, 𝑀1 is the growth in the money supply or the monetary items (excludes reserves) available in the economy, and 𝐶𝑃𝐼 is the growth in the Consumer Price Index of the specific country, respectively. The coefficients are corrected for the presence of heteroskedasticity and autocorrelation with the Breusch-Godfrey test. The standard errors appear in parentheses, where ***,** and * correspond to 1%, 5%, and 10% significance level, respectively.
FTSE DAX IBEX AEX CAC SMI
22 dividend yield paid out in the previous month is positively interpreted by the stock market. The ability to predict the MRP with the P/E ratio is not significant. There is no statistical evidence to reject hypothesis 4, which indicate that there is no relation between both variables. This finding is in contrast to previous empirical literature who posit that high 𝑃/𝐸 ratios are associated with the overvaluation of the aggregated stock market (Hirshleifer et al. 2008). This overvaluation leads to lower subsequent returns. 𝑉𝑂𝐿 remains an important determinant of the MRP and the sign of the relationship do not change. The conclusion and interpretation about the other determinants do not change.
Several authors find compelling evidence of the ability to predict the MRP with the explanatory variables on a long time horizon (Jacquier et al., 2005; Bätje and Menkhof, 2013; Neely
et al., 2014). The data is therefore converted into an quarterly frequency with the econometric
software EViews. The regression model on a quarterly basis is presented in Appendix I. The results indicate that the predictability of the MRP is very weak at quarterly horizons, which is in contrast to the results of Bollerslev et al. (2009). The authors find the largest degree of predictability at quarterly horizons.
23 Table 7
Lagged average dividend yield and price/earnings ratios on an annual basis
This table shows the results of the simple bivariate analysis based on annual data to find the determinants of the MRP in the period 1995 - 201419. 𝐷𝑌 (-1) and 𝐷𝑌 (-2) is measured as the average dividend yield of the market index with one and two time lags, 𝑃/𝐸 (-1) and 𝑃/𝐸 (-2) is the average price/earnings ratio with one and two time lags, respectively. The coefficients are corrected for heteroskedasticity with the white test. The standard errors appear in parentheses, where ***,** and * correspond to 1%, 5%, and 10% significance level, respectively.
FTSE DAX IBEX AEX CAC SMI
𝐷𝑌 (-1) 0.0156*** 0.0046 0.0017 0.0067 0.0055 0.0033 (0.0047) (0.0059) (0.0032) (0.0054) (0.0056) (0.0053) 𝑅2 0.3916 0.0347 0.0171 0.0823 0.0537 0.0224 𝐷𝑌 (-2) 0.0118** 0.0046 0.0024 0.0048 0.0048 0.0052 (0.0055) (0.0062) (0.0032) (0.0056) (0.0057) (0.0056) 𝑅2 0.2238 0.0331 0.0333 0.0437 0.0422 0.0511 𝑃/𝐸 (-1) -0.0011 -0.0015 -0.0008 -0.0007 -0.0013 0.0000 (0.0008) (0.0012) (0.0010) (0.0009) (0.0013) (0.0012) 𝑅2 0.1074 0.0821 0.0340 0.0294 0.0577 0.0000 𝑃/𝐸 (-2) -0.0012 -0.0022* -0.0016* -0.0012 -0.0018* -0.0012 (0.0008) (0.0012) (0.0009) (0.0009) (0.0013) (0.0012) 𝑅2 0.1305 0.1770 0.1667 0.0986 0.1029 0.0548 4.2 Further evidence
Hedi and Fredj (2010) emphasizes that the crisis has significant impact on the magnitude of the MRP. Table 8 shows the differences in the magnitude of the MRP on a monthly basis in several states of the economy. The mean MRP, number of observations and t-statistics are also presented in table 8. I applied a one-tailed independent sample t-test. Panel A presents the differences in the magnitude of the MRP in the pre-crisis and post-crisis. Although the MRP in the pre-crisis is higher than the post-crisis, the t-statistics are not significant across indices. There is no rejection of hypothesis 8 and therefore no evidence of a higher MRP in the pre-crisis. Panel B shows that the null hypothesis can be rejected if the differences are tested between the pre-crisis period and the global financial crisis in 2007 – 2008. The MRP is significantly lower in the global crisis due to the lower or even negative market returns of the stock indices. The negative MRP implies that the risk-free rate earns higher return than the stock market. The reaction of the shareholders to invest in bond
19 A simple regression instead of a multiple regression is conducted, since there only 20 observations on an
24 Table 8
Differences in the magnitude of the MRP
This table shows the differences in the magnitude of the MRP on a monthly basis in several states of the economy or the market. Panel A refers to differences in the magnitude of the MRP in the pre- and post-crisis, which correspond to the period 1995 – 2007m07 and 2007m08 - 2014, respectively. Panel B represents the differences in the pre-crisis and the crisis (2007m08 – 2008). Panel C represents the differences between the mean individual MRP per market index and the mean MRP of Europe, which is measured with the Eurofirst100. Panel D presents the differences in the MRP in bull and bear markets20. A one-tailed independent sample t-test is applied to test for the differences between the samples, where ***,** and * correspond to 1%, 5%, and 10% significance level, respectively.
FTSE DAX IBEX AEX CAC SWISS
Panel A: Differences pre- and post-crisis
Mean MRP pre-crisis 0.0053 0.0064 0.0096 0.0069 0.0071 0.0072 Observations 151 151 151 151 151 151 Mean MRP post-crisis 0.0027 0.0032 0.0004 0.0008 0.0000 0.0015 Observations 89 89 89 89 89 89 T-statistic 0.3916 0.3471 0.9792 0.6682 0.8220 0.7567
Panel B: Differences pre-crisis and crisis
Mean MRP pre-crisis 0.0053 0.0064 0.0096 0.0069 0.0071 0.0072 Observations 151 151 151 151 151 151 Mean MRP in crisis -0.0258 -0.0346 -0.0298 -0.0472 -0.0377 -0.0287 Observations 17 17 17 17 17 17 T-statistic 1.8839** 2.2296** 2.3504** 2.2631** 2.3702** 2.3393**
Panel C: Differences individual MRP and mean MRP of Europe Mean individual MRP 0.0042 0.0046 0.0060 0.0041 0.0039 0.0051 Observations 240 240 240 240 240 240 Mean MRP Europe 0.0032 0.0032 0.0032 0.0032 0.0032 0.0032 Observations 240 240 240 240 240 240 T-statistic -0.1307 0.0435 0.2093 -0.0569 -0.0928 0.0279
Panel D: Differences in bull and bear markets
Mean MRP peaks 0.0047 0.0097 0.0102 - 0.0091 0.0107 Observations 202 200 174 - 199 183 Mean MRP troughs 0.0013 -0.0281 -0.0050 - -0.0211 -0.0071 Observations 38 40 66 - 41 44 T-statistic 0.2764 2.4303*** 1.3706* - 2.2555** 1.9413**
20 Unfortunately, the ECRI provides no information about the peaks and troughs in the Netherlands. Therefore,
25 markets is also called ‘’the flight to quality’’. The government normally intervenes to this situation by spending a huge amount of capital and thereby cutting the risk-free rate until the MRP turns positive.
The individual MRPs within six OECD countries and the mean MRP in Europe are compared to test whether the MRP in major countries differs from the mean MRP in Europe. The mean MRP in Europe is measured with the Eurofirst100 and represents the stock market index of Europe. The t-statistics in panel C indicate that there is no statistical difference between the individual MRPs in major European countries and the mean MRP in Europe. The finding is consistent to the results of the Absolute Research Strategy, who also find that the MRP in major European countries are almost equal to the MRP of Europe. Panel D presents the differences in the magnitude of the MRP in bull and bear markets. Consistent with Bätje and Menkhoff (2013), the amplitude of bull markets is longer than bear markets. The findings imply that the MRP is the highest at the top of the bull market and the lowest at the bottom of bear markets. The results are robust across countries (except for FTSE). There is statistical evidence to reject hypothesis 10. An upward market trend increases the MRP as the return on the market indices increases.
The findings of the multiple regression analysis to examine the relation between the MRP and the explanatory variables in the pre-crisis and the post crisis are represented in table 9. The results indicate that the relation between the determinants and the MRP are not different before and after the financial crisis. The adjusted 𝑅2 is also almost the same. Although McQueen and Roley (1993) find that the relation of the 𝑀1 depends on the state of the economy, the results of table 9 are not able to confirm their findings.
26 Table 9
Determinants of the Market risk premium in the pre- crisis and post-crisis
This table shows the results of the multiple regression analysis based on monthly data to find the determinants of the MRP in the period 1995 – 2007m07 and 2007m08 – 2014. The pre- crisis and post-crisis correspond to model (1) and (2), respectively. The regression for both sub-samples is as follows:
𝑀𝑅𝑃𝑡= 𝛼0+ 𝛽1𝑉𝑂𝐿𝑡+ 𝛽2𝐷𝑌𝑡−1 + 𝛽3𝑃/𝐸𝑡−1+ 𝛽4𝐺𝐷𝑃𝑡+ 𝛽5𝑀1𝑡 + 𝛽6𝐶𝑃𝐼𝑡 +ut
where, 𝑉𝑂𝐿 is the monthly historical volatility of the specific market index, 𝐷𝑌 is the average dividend yield of the market index, 𝑃/𝐸 is measured as the average price/earnings ratio of the market index, 𝐺𝐷𝑃 is the growth in Gross Domestic Product of the country where the market index is located, 𝑀1 is the growth in the money supply or the monetary items (excludes reserves) available in the economy, and 𝐶𝑃𝐼 is the growth in the Consumer Price Index of the specific country, respectively. The coefficients are corrected for the presence of heteroskedasticity and autocorrelation with the Breusch-Godfrey test. The standard errors appear in parentheses, where ***,** and * correspond to 1%, 5%, and 10% significance level, respectively.
FTSE DAX IBEX AEX CAC SMI
27 4.4 Robustness test
Another proxy for the market risk is the implied volatility. The implied volatility measures the expected volatility of the market in the following 30 calendar days by analyzing call and put option prices. The implied volatility of four market indices are available from 2000 and computed from Datastream21. A simple bivariate regression model is conducted to measure the impact of the implied volatility on the MRP. The formula is as follows:
𝑀𝑅𝑃𝑡 = 𝛼 + 𝛽𝐼. 𝑉𝑜𝑙𝑡+ u𝑡 (10)
where, 𝐼. 𝑉𝑜𝑙𝑡 is the implied volatility of the market index in month t. The simple regression model
of the historical volatility with the same time horizon as the implied volatility are also presented in Appendix K. The results suggest a negative contemporaneous relation between the implied volatility and the MRP. The negative relation is consistent with the findings of Giot (2005), who examines the VIX and compares the variable with the stock returns of the S&P 500. Giot (2005) also find an asymmetric effect, where the negative returns are associated with a greater change in the implied volatility compared to positive returns22. The implied volatility shows large peaks in 2007 -2008, which coincides with the global financial crisis. The Levene’s test assesses the variances of the two samples and makes no assumption about the normality of the distributions and is more robust than the F-test. The formula is presented in the paper of Brown and Forsythe (1974) on page 364. The Levene’s test statistic is also presented between brackets in Appendix K. The results indicate a significant difference between the coefficients of the historical volatility and the implied volatility. The multiple regression analysis is conducted to test if the differences in proxies hamper the results. The results are presented in table 10 and indicate that the lagged 𝐷𝑌 performs stronger than earlier regressions. The results show no major changes in the significance levels of the other explanatory variables and the adjusted 𝑅2.
Another robustness test is to remove the financial crisis of 2007 – 2008, because the results could be noisy. The findings are available on request and show no major changes in the determinants of the MRP. The findings indicate that the global financial crisis do not hamper the results.
Several authors use the local 3-month treasury bills as a proxy for the risk-free rate (Dimson
et al., 2008). A one-tailed independent sample t-test is applied to test if the 3-month German
treasury bill is significantly lower than the 3-month local treasury bills. The results are presented in
21 Unfortunately, not all implied volatilities for each market index is available in DataStream. The four market
indices are FTSE 100 implied volatility index, DAX volatility index, AEX volatility index and CAC40 volatility index, respectively.
28 Table 10
Different proxies for the market risk
This table shows the results of the multiple regression analysis to test if there are significant differences in the determinants of the market risk premium if we differentiate between the historical and implied volatility23. The regression is as follows:
𝑀𝑅𝑃𝑡 = 𝛼0+ 𝛽1𝐻. 𝑉𝑂𝐿𝑡+ 𝛽2𝐷𝑌𝑡−1 + 𝛽3𝑃/𝐸𝑡−1+ ut (1) 𝑀𝑅𝑃𝑡 = 𝛼0+ 𝛽1𝐼. 𝑉𝑂𝐿𝑡+ 𝛽2𝐷𝑌𝑡−1 + 𝛽3𝑃/𝐸𝑡−1+ ut (2)
𝐻. 𝑉𝑂𝐿 is the monthly historical volatility of the specific market index, 𝐼. 𝑉𝑂𝐿 is the implied volatility and measures the expected volatility of the market in the following 30 calendar days, 𝐷𝑌(-1) is the average dividend yield of the market index with one time lag, and 𝑃/𝐸(-1) is the average price/earnings ratio of the market index with one time lag, respectively. The coefficients are corrected for the presence of heteroskedasticity and autocorrelation with the Breusch-Godfrey test. The standard errors appear in parentheses, where ***,** and * correspond to 1%, 5%, and 10% significance level, respectively.
table 11. The insignificant t-statistics indicate that we are indifferent between the local treasury bills and the German treasury bills as a proxy for the risk-free rate.
23
Note that I only test for the most important factors determining the MRP, since previous empirical literature and/or table 5 and 6 indicate that these factors predominantly determine the MRP. The factors are the 𝑉𝑂𝐿, lagged 𝐷𝑌 and lagged 𝑃/𝐸 ratio,
FTSE DAX AEX CAC
29 Table 11
Relation between the local treasury bills and the German treasury bill
This table shows the differences in the coefficients of the local treasury bills and the German treasury bills. The independent sample t-test is applied to test for a difference between the 3-month German treasury bills and the 3-month local treasury bills. The mean risk-free rate of the German is 0.0020.
UK Spain France Switzerland
Mean risk-free rate 0.0031 0.0025 0.0021 0.0013
Obs. 240 240 240 240
T-statistic 0.3027 0.3037 0.3959 -0.6844
5. Conclusion
The MRP is crucial for valuation and portfolio theory, as investors estimate the cost of equity and determine the optimal allocation of assets with the key valuation parameter. This study is an attempt to elucidate the determinants of the MPR in 1995 – 2014. Several macroeconomic variables and financial ratios are hypothesized to explain the dispersion in the MRP. Previous empirical literature demonstrates that financial ratios influence the magnitude of the MRP and are better able to capture the business conditions of the market (Campbell and Driebold, 2009; Dangl and Halling, 2012). By investigating the relations of the determinants of the MRP and distinguishinges between business cycles, this paper has added to the broad available studies. This study confirms some of the determinants which influences the magnitude of the MRP.
30 The sample is split up into two sub samples to distinguish between the pre-crisis and post-crisis effect on the magnitude of the MRP. The difference is not significant. There is also no evidence to suggest that estimating the MRP when differentiating between the sub samples, enhances the forecasting ability. Furthermore, this research also examines the MRP in bull and bear markets and find that the magnitude of the MRP is higher in the former. However, the relation between the MRP and the determinants do not change when differentiating between bull and bear markets.
A model including time lags is conducted to examine the ability to predict the MRP in the next month or year. I find weak evidence to forecast the MRP on a short and long time horizon. The former is consistent with previous literature, while the latter contradicts literature. The ability to predict the MRP with the average P/E ratio is not significant. However, the average dividend yield is able to predict the MRP on a short horizon and robust across most indices The positive relation implies that a high 𝐷𝑌 paid out in the previous month is interpreted by the market as a positive sign for the next month. The finding is in contrast with the Dividend Irrelevance theory of Modigliani and Miller (1961) and the Efficient Market Hypothesis. Inferences based on a long time horizon are interpreted with caution, because of the relatively short time period of data.
The results have important implications for the strategic allocation of assets, because shifting the asset allocation for changing MRPs increases the probability of higher expected returns. It is preferable for investors to increase the allocation to stocks when the 𝐷𝑌 are expected to grow, due to the significant positive relation (Hewitt, 2015). If the explanatory variable change and the MRP decreases, than subsequently the cost of capital decreases, which results in more investments in the economy and higher economic growth. In summary, financial professional should incorporate estimates of the MRP that reflect current market conditions.
Although this paper provided key insights on the determinants of the MRP, the research did not measure the risk aversion of the investors. However, Dangl and Halling (2012) find an increase (decrease) in the MRP during an economic recession (expansion) due to the changes in the risk aversion of the investors. Future research in this topic is necessary and should examine the impact of the time-varying risk aversion of the investors on the MRP. In addition, a longer time period of data is recommended as it yield insights into the stability of the parameters and make the forecast more reliable and accurate. Further research can also model the MRP out-of-sample to test the forecasting ability of the model using extrapolation. The out-of-sample regression is extremely important for financial professionals, because future values are not known ex-ante.
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