Models of expected European equity returns incorporating implied volatility and a simple measure of investor sentiment

Transcription

1 TILBURG UNIVERSITY Models of expected European equity returns incorporating implied volatility and a simple measure of investor sentiment Ioannis Laliotis Department of Finance 11/17/2013

2 Abstract This thesis aims to create simple, intuitive models for the estimation of the short and long-term expected rates of return on European equity. To do so, it closely follows the methodology of Doran, Ronn and Goldberg (2005) A Simple Model for Time-Varying Expected Returns on the S&P 500 Index. In the process of creating the models, two forward looking measures are used: the forecasted growth rate of the economy and the implied volatility from the options market. By combining existing models of expected returns that incorporate these measures, a forward looking Market Price of Risk is created. This measure is then used to test the following hypothesis: that investors have a varying risk aversion that is affected by the recent observed realized returns of the representative equity index. Indeed such a relationship proves statistically significant and with a high R 2. The coefficients resulting from the regressions (that were run to test the hypothesis) are then used to form new models of expected return. The results of these models are examined and also compared with realized returns over the time period. 1

3 1. Introduction Expected rates of return are a vital component in almost every model found in finance. They are used to estimate the cost of capital in corporate finance and as a discount rate in security valuation. The idea that risk affects investment decisions is logical and also central in risk and return models in Finance. Riskier investments should have higher expected returns than safer investments in order for investors to consider them. Therefore the expected return of any investment can be written as the sum of the risk-free rate and a risk premium to compensate for the risk-taking. The challenge, in both financial theory and practice, is on how to quantify the risk in an investment so that the resulting expected return fairly compensates for risk. A central measure used towards this goal is the premium that investors demand for investing in the average or benchmark equity investment (or for investing in equities as a class), the equity risk premium. The broad goal of this thesis is to add to the literature examining the equity risk premium and consequently the expected rate of return of equity. More specifically, it aims to use and produce models that incorporate ex-ante variables. A common drawback of some of the most traditional models is that they are using past measures. For example, CAPM as well as any of its multifactor extensions use ex-post measures, such as historical volatility, historical returns etc. In this thesis ex-post measures are in fact used, but they are combined with ex-ante measures to produce estimates that can be termed as more forward-looking. During my research on the topic, I noticed a paper that is a good fit with this idea and became my major source of inspiration. Doran, Ronn and Goldberg (2005) A Simple Model for Time- Varying Expected Returns on the S&P 500 Index. Obviously, the S&P 500 Index is the typical equity asset benchmark. The paper creates time-varying models of expected return that use concepts from established financial theory but are also intuitive in their application. The paper estimates a parametric form of the Market Price of Risk, also named the Sharpe Ratio, of the S&P 500 Index. This estimate incorporates two forward-looking measures: The economy s growth rate forecast and the option market s implied volatility on the equity index. This estimate is then used to test the following hypothesis: that investors have a time-varying risk aversion. The hypothesis, which is confirmed by the data, is that investors are willing to take more risk per 2

4 unit of return when realized wealth levels are high. The proxy for realized wealth levels are the capital gains of an investor that had invested in the index between 5 and 6 years ago and sold the index at present day. This measure is also called as an investment sentiment in the Doran, Ronn, Goldberg paper; this thesis simply refers to it as observed past returns. The paper presents several models of return; all of them are variations of the same main concept. Investors accept a certain forward looking and time varying Sharpe ratio. This Sharpe ratio is multiplied with the implied volatility to provide an expected excess return which is then added to the risk free rate to provide the expected total return. The Sharpe ratio is central to the models. It is decomposed further by testing for a relationship between the Sharpe ratio and past returns of the index in recent years. The past capital returns on the index are assumed to be observed by the representative investor; they are assumed to be a simple proxy for investor sentiment as mentioned above. After the relationship between observed realized returns and the forwardlooking Sharpe ratio is quantified, this relationship is incorporated into the models. Since the models are conditional on the implied volatility they are forward-looking and timevarying. Implied volatility is the volatility implied by option pricing formulas and is often used as an estimate of next period volatility by investors. On the other hand, historic volatility is based on the variation of past realized returns and is often not a reliable predictor of future volatility; historical volatility is not used in this thesis. The models presented and examined herein are identical to the models used in the Doran, Ronn and Goldberg (2005) paper. However there are two differences in the data used that make the findings of this thesis relevant: - The models in this thesis use European data. My source of inspiration uses US data. Overall, in financial research, findings in the US market do not always match findings in the European market. - The data span from 1999 until Doran, Ronn and Goldberg examine 1986 until It is interesting to determine if the same pattern of expected equity premium appears in more recent times. The thesis is structured in the following way: 3

5 Chapter 2 gives an overview of previous literature on the topic of expected returns, the equity premium and the equity premium puzzle. Chapter 3 describes the models. For each model the methodology and the results are presented in succession. The methodology involves constructing the forward-looking Market Price of Risk and then describing the hypothesis that this Market Price of Risk is affected by past observed capital returns. OLS regressions are used to test the relationship and the results section for each model contains the results of those regressions. Chapter 4 describes the data used to construct the models, such as the dividends and the growth rate forecasts. Chapter 5 takes into account the relationship between the Market Price of Risk confirmed in chapter 4 and constructs new models of expected returns. The model results, should they have been used during the time period are examined and also compared with the realized returns, both in average values and correlations. By doing so, it is also tested if the US equity premium puzzle (expected returns being significantly lower than realized returns) is also found in European data using these models. Chapter 6 provides in short the conclusion of the thesis. 2. Literature The equity risk premium is a central component of every risk and return model in finance. It is used as an important input in a variety of finance applications, ranging from valuation to asset allocation to corporate finance. However, finance practitioners and academics have not reached a consensus on the estimation of the risk premium, sprouting extensive research and literature on the topic. 2.1 Factors that affect the equity premium Before the estimation itself, it is important to examine the possible factors that determine the equity risk premium. Perhaps the most fundamental underlying factor is the risk aversion of the investors in the market. Risk aversion arises from the fact that equity carries risk compared to the 4

6 risk free asset ; as the average investor risk aversion increases so does the market risk premium. Proof of the risk aversion effect can be found in the fact that there is substantial evidence of individuals becoming more risk averse as they get older. The logical conclusion of this evidence is that markets with older investors in total should have higher risk premiums than markets with younger investors, for any given level of risk. Bakshi and Chen (1994) examined risk premiums in the United States and noted an increase in risk premiums as investors aged. Liu, Z. and M.M. Siegel (2011) have similar findings. Risk aversion research also led to the equity risk premium puzzle. Mehra and Prescott (1985) observed historical risk premiums (which they estimated at about 6% at the time of their analysis) were too high, and that investors would need implausibly high risk-aversion coefficients to require these premiums. Since the equity risk premium is one of the concerns of this thesis, related literature for this topic is examined in more detail on a later section. Economic risk is another important determinant that arises from more general concerns about the health and predictability of the overall economy. The market risk premium should be lower in an economy with predictable inflation, interest rates and economic growth than in one where these variables are volatile. Lettau, Ludwigson and Wachter (2007) link the changing equity risk premiums in the United States to shifting volatility in the real economy. Information also determines part of the market risk premium. When investors invest in equities the risk in the underlying economy is manifested in volatility in the earnings and cash flows reported by individual firms in that economy. Yee (2006) defines earnings quality in terms of volatility of future earnings and argues that equity risk premiums should increase (decrease) as earnings quality decreases (increases). Moreover, Lau, Ng and Zhang (2011) look at time series variation in risk premiums in 41 countries and conclude that countries with more information disclosure, measured using a variety of proxies, have less volatile risk premiums and that the this information is made more important during crises. In addition to the risk from the underlying real economy and imprecise information from firms, equity investors also have to consider the additional risk created by illiquidity. If investors have to accept large discounts on estimated value or pay high transactions costs to liquidate equity positions, they will be pay less for equities today (and thus demand a large risk premium). 5

7 Gibson and Mougeot (2002) look at U.S. stock returns from 1973 to 1997 and conclude that liquidity accounts for a significant component of the overall equity risk premium, and that its effect varies over time. Baekart, Harvey and Lundblad (2006) present evidence that the differences in equity returns (and risk premiums) across emerging markets can be partially explained by differences in liquidity across the markets. 2.2 Estimation methods Finance literature also diverges in the methodology of estimating the market risk premium. There are three major approaches in estimating the Market Risk Premium: Historical premium, the most commonly used method, uses the difference between the past returns of equities minus the return on a risk free asset (usually a government bond). Despite the simplicity and the widespread use of this approach and the fact that common historical data are available to researchers and practitioners, there is a noticeable difference in the market risk premia calculated. The main reasons for historical premia deviations include: differences in time period of the data, differences in the equity indices and the corresponding risk free benchmark selected and differences between the way returns are averaged over time. One other factor affecting the historical premium calculation is the survivorship bias. The return on equity is equaled to the return of a broad equity index that is market value weighted. This means that stocks of companies that go bankrupt or severely underperform will be removed from the index. That leads to an inflated equity premium based on the index since it leaves out losers in the stock market. Jorion, Philippe and William N. Goetzmann (1999) estimated that the survivor bias added 1.5% to the US equity risk premium. Survey premium on the other hand uses surveys of investors, managers and academics to gauge future expectations of the MRP. Although survey technology increases, investor predictions are very sensitive to sentiments about the market, resulting in limited value of predictive power. For example Fisher and Statman (2000) conclude that there is a negative relationship between investor sentiment (individual and institutional) and stock returns. Graham, J.R. and C.R. Harvey, (2010) have focused on CFO expected market risk premiums and reported a 3% premium at the June 2010 survey. Welch (2000) surveyed 226 finance academics. 6

8 Forward looking or Implied market premiums. The drawback of any historical premium is that it is backward looking. The objective of investors about to take risk by investing in equity is to estimate an updated, forward-looking premium; it is counter-intuitive to trust in mean reversion and past data. Additionally problems of historical data is that during stressed markets the historical premium decreases (because of bad performance of equity) which is counter to what investors will require of an equity risk premium, now that the market is more risky. The limitations of the historical risk premium, which makes a forward looking premium estimation necessary, is what captured my interest. There are three approaches in finance literature for estimating equity risk premiums that are forward looking: DCF Model Based Premiums also used by Fama, E. F., and K. R. French. (1988). Default Spread Based Equity Risk Premiums where the spread between the interest rate on corporate bonds and the treasury bond rate, is used as the risk premium. Option Pricing Model based Equity Risk Premium. This school of estimation is where the models presented in this thesis belong to. Santa-Clara and Yan (2010) use options on the S&P 500 to estimate the ex-ante risk assessed by investors from 1996 and 2002 and back out an implied equity risk premium on that basis. This thesis closely follows the logic and methodology of Doran, Ronn and Goldberg (2005), A Simple Model for Time-Varying Expected Returns on the S&P 500 Index. It also belongs under the Option Pricing Model based Equity Risk Premium category. Doran, Ronn and Goldberg decompose two fundamental models for valuing stocks into certain parts. Then they quantify the ex-ante market price of risk formed out of this combination and explain the time-variation in the market price of risk with a simple measure of investor sentiment. This measure is found to be negatively related to the premium, such that the higher the perceived wealth, the lower the market price of risk. Using that measure of investor sentiment, they are able to explain as much as 50% of the variation in expected returns; therefore indicating potential high and low seasons in the market. 2.3 Equity risk premium literature leading to the premium puzzle One of the pivotal pieces of finance literature concerning the market risk premium is the CAPM model of Sharpe (1964) and Lintner (1965). These authors derived the relationship between equity returns and a market-wide factor of risk and consequently sparked an extensive theoretical 7

9 and empirical research. One of the most examined aspects of the model is the notion of the equity risk premium as well as the market price of risk. This measure of reward to variability ratio was used by Sharpe (1966) to describe mutual funds; it is often called as the Sharpe ratio or measure. Of particular interest, as it is most relevant to the subject of this paper, is the ex-ante Sharpe Ratio, introduced by Sharpe (1994). The purpose of ex-ante Sharpe ratio is to measure expected returns (as opposed to ex-post realized returns that the typical Sharpe ratio measures). Consequently, there are two important topics to be examined. Which are the factors that affect the amount of the equity risk premium and an estimation of the equity risk premium itself. According to the CAPM model, used as a staple in financial theory, there is positive relationship between the level of volatility and the amount of the risk premium; but that is occasionally disputed by empirical evidence. Campbell (1987) as well as Glosten, Jagannathan and Runkle (1993) report a negative relationship between conditional volatility and the equity risk premium, contrary to CAPM theory. On the other hand, Harvey (1989) as well as Turner, Startz and Nelson (1989) report a positive relationship. An explanation of the negative relationship found by Campbell (1987) and Glosten et al. (1993) was managed by Scruggs (1998) who split the CAPM model into a partial relation in a two-stage estimation. Brandt and Zhou (2004) attempt to tackle the differences mentioned above by implementing a VAR technique. They conclude that these differences can be explained by the conditional and unconditional correlations and by incorporating time-varying volatility. Of further interest concerning time-varying expected returns are the volatility ratio tests of LeRoy and Porter (1981) as well as and the long-horizon autoregressions of Fama and French (1988a, b). Campbell and Viceria (2005) further examine the time variation in expected returns. They suggest that investors, especially aggressive investors, plan to benefit from market-timing (or tactical asset allocation); their aim is to maximize short-term returns, based on the predictions of their forecasting models. As always, trying to predict asset returns is a haphazard process. P astor and Stambaugh (2001) focused on this principal and concluded it is difficult to set-up the optimal market timing strategy. 8

10 Ferson and Harvey (1991b) used a multi-beta asset pricing model which takes into account risk exposure to the market as well as the interest rate and inflation to explain realized returns. Lewellen (1999, 2004) has instead incorporated explanatory variables such as the dividend yield, short rate, term premium, book-to-market, and the default premium. The results of Lewellen are put in question by Boudoukh and Richardson (1993), Stambaugh (1999), and Ferson et al. (2003), citing statistical issues. Proving even further the difficulties in estimating the expected equity premium, Mehra and Prescott (1985) introduced the equity premium puzzle. They report that the annualized rate of return on stocks in excess of the risk-free rate is higher than can be explained by the classical financial theories by about 6.8%. There has been further research into the equity premium puzzle. Mehra (2003) focused on the fundamental pricing relationship. He concluded that the growth rate of consumption does not have enough variation to explain the observed high equity premium. Mehra observed that using high levels of risk aversion (also an important factor in most financial models) results in risk premiums that are too low and consequently risk-free rates that are too high. However he believes this is a quantitative issue and that current theory should not be dismissed. Other attempts to explain the equity premium puzzle took into account several factors commonly found in finance literature, such as: altering preferences, survivorship bias, incomplete markets and dismissing rare events. Despite previous research, there is no clear explanation to the equity premium puzzle. The puzzle is one of the main motives behind this thesis. Doran, Ronn and Goldberg (2005), whose work this thesis replicates, examine the equity premium puzzle by contrasting the results of the produced expected return models with realized returns. Unfortunately, like Mehra and Prescott (1985) findings, the predicted value of the expected return is lower than realized returns over the evaluation period. Similar to the findings of Fama French (2002) the expected return is about 4% lower than realized returns. 9

11 3. The models The models presented and examined herein are identical to the models used in the Doran, Ronn and Goldberg (2005) paper, which I will occasionally refer to as DRG for short. The models are a combination of the Gordon-Shaprio (1956) Williams (1938) dividend-growth model (also known as dividend discount model) and the Sharpe-Lintner security market line. The differences in each model arise from different interpretations of (a) the growth rate(s) and (b) from the different investment horizons (short-term and long-term). The models and the measures used for each models are presented below, in ascending complexity: Methodology - Short-Term Expected Return Model (Model 1) Initially, I combine two fundamental models used to estimate the expected equity returns. The first one to be used in the combination is the dividend discount model:, where is the dividend one year ahead from time t. It can be rearranged to: (1a) under the basic assumption that. This assumption essentially means that the current dividend D 0t will grow at the growth rate g 1t for one year. The definition of each variable appears below: µ t = Expected/required rate of return on the equity asset as of date t. In this thesis the equity asset is the FTSEUR1ST 300, a broad European equity index. P t = Price of equity asset at date t. D 0t = Dividends paid over the past year. g 1t = One-year dividend growth rate as of date t and capital-gains forecast over the next twelve months. It is important to note that the growth rates used are forecasts and not realized growth rates. 10

12 As mentioned in greater detail in the Data chapter, the dividend growth rates are assumed to be equal to the European GDP growth-rate forecast. Since the FTSEUR1ST 300 is a broad index, its dividends and capital gains are assumed to grow at the rate of the overall European economy; this is the rationale behind using the European GDP growth rate. It is important to note that each g 1t is retrieved from a time-series of forecasts; not realized GDP growths rates. This is important to make the models forward looking and is also the method used by my source of inspiration. The second model to be used in the combination is: (2a) where: = Market Price of Risk, or Sharpe Ratio, as of date t r 1t = Short term rate of interest as of date t σ t = Equity asset s annualized volatility, as of date t. In order to make the model forward looking, the traditionally used volatility of realized returns is substituted with a future volatility estimate. The volatility estimate used in this thesis is the implied volatility derived from an option pricing model. Those implied volatilities can be found under the VSTOXX volatility index; more information about the index appears under the data section. Therefore σ t =VSTOXX t or for short VST t. This model uses the quintessential logic of expected returns in its simplest form. The expected return is comprised of the return of the risk free asset plus the equity premium the investor requires. λ t is the Market Price of Risk the investor has determined as fair. It is not a fixed amount as the traditional Markowitz theory would imply by assigning fixed risk aversions to investors. Instead in can change over time based on investor sentiment. The main hypothesis of this thesis is that this investor sentiment is affected by the recent realized returns of the index. This λ t is then multiplied by the implied volatility of the market to produce the equity premium. By equating (1a) with (2a) I end up with which I solve for the ex-ante Sharpe ratio measure: (3a) 11

13 3a is the previously mentioned forward looking Sharpe ratio. It is the most important relationship in the thesis (and varies slightly in the next models). The right hand side of 3a is based on available data; λ t is an observable variable Methodology- Regressions for Model 1 Using the data above I now have a time series of λ t. The idea that I intend to test in this thesis, in general terms, is that this forward-looking Market Price of risk is affected by the certain variables. The test in this case is that the Market Price of Risk (or Sharpe Ratio) accepted by investors is affected by the observed price changes of the index. The extension of this is that investors do not have a static risk aversion; they are willing to undertake more or less risk depending on the performance of the index that they observe. Another way to consider this behavior, is to assume that an investor who holds an equity asset and observes a positive past realized return feels wealthier and is willing to undertake more (or less) risk in the future. To put it in equation form, the hypothesis assumes that λ t can be further decomposed such as λ t =λ 0 +λ 1 χ t (where χ t is the observed past returns of the index). This past observed realized return, a proxy of investor sentiment, is measured by the current price of the index relative to a price of the index in the past; nominated. It is argued in my source of inspiration, DRG that investors have a limited memory. This can be due to the fact that investors consider older data are not relevant for tomorrow s prediction or that investors have a limited attention. Therefore the time T used should be relatively short. Similar to the source paper, I am using two measures for the past price. One is simply the price of the index exactly 5 years ago, nominated as P t-5. The second is the average price of the index between 6 years ago and 5 years ago, nominated P t-5,6. P t-5,6 is in other words, the average price in the previous 12 months from the point of view t-5. The second measure is expected to perform better, since it avoids irregularities. For example if the price is particularly low at exactly 5 years ago (possibly due to a market shock or other irregularity) it will drive the observed capital gain return abnormally high. The averaged measure makes sure that these irregularities do not affect the measure much. Both measures are used in the first model. The one with the higher explanatory power is determined as the better fit and used exclusively for the next models. 12

14 The hypothesis for this model has two variations. One tests if there is a linear relation between the Market Price of Risk λ t and the observed price change while the second tests for a quadratic relation. Therefore the regression specifications are the following: { where χ t = is alternatively P t /P t-5 and P t /P t-5,6 where P t-5,6 is the average value of the FTSE index over the period between 6 years ago and 5 years ago as described above Results - Regressions, Linear Model 1 Model 1 Linear OLS Regression results ECB 1 Year Ahead Reuters 1 Year Ahead ECB Current Year χ t = P t /P t-5 χ t = P t /P t-5,6 χ t = P t /P t-5 χ t = P t /P t-5,6 χ t = P t /P t-5 χ t = P t /P t-5,6 λ t-stat (-13.04) (-13.36) (-6.34) (-7.16) (-9.28) (-9.79) λ t-stat (25.68) (26.05) (15.1) (16) (18.6) (19.19) R Obser T- statistics are in parenthesis. The numbers of observations appear at the last row. 13

15 In the first two columns the dependent variable λ t is the one calculated by using the ECB forecast for the growth rate. In the second pair of columns the growth rate is taken from the Reuters Poll forecast and the final pair of columns is based on the current year forecast instead of the one year ahead. Although there are differences in explanatory power among specification results, the common fact is that λ 1 is negative. The negative sign of λ 1 reflects that as the observed capital gain on the index increases, investors believe in a greater opportunity for future wealth; consequently their required compensation per unit of risk declines. It could also mean that investors that already have invested in equity have now more funds and this causes them to accept more risk. Beside the coefficient of the price change, λ 0 (the constant) is both economically and statistically significant. It means that there is a constant component in the risk aversion of each investor and it has a significant impact on the required market price of risk. These finding are in line with my major source of inspiration, DRG. Using US data, they also find a negative sign for λ 1 and a positive sign for λ 0. The relation in magnitude between the coefficients is also nearly identical. The constant λ 0 is for the P t /P t-5,6 regression while λ 1 is nearly three times lower at (in absolute value). In the same case, in European data used by this thesis, λ 0 is while λ 1 stands at ; also nearly three times lower. The main difference between European and US data is the magnitude of the coefficients; US coefficients are higher overall. This occurs because the dependent variable is smaller for European data; mainly due to higher equity returns in the US market (both expected and realized) that has been numerously documented. Additionally, the time series used in this thesis also include the years of the recent financial crisis, while in DRG available data end at It is interesting to note that expected premium actually increases after While the overall return decreases, especially during the very low forecasted growth rates at the end of 2011 and onwards, the expected premium increases due to the dramatic drop of the government treasury yields during the same period. Essentially, the low fixed income returns, used as a risk free rate, result in equity investments retaining their attractiveness during the crisis years. 14

16 It is important to also see which specification across the model performs better. The first comparison is across the different forecast sources. It is apparent that the ECB 1 year forecast has the higher explanatory power overall. It has results with higher statistical significance in the depend variables but more importantly has the highest R 2 by a significant margin. The Reuters forecast fairs much worse in comparison, to both ECB forecasts. Reuters R 2 is less than half of ECB 1 year ahead forecast. It indicates that past returns have a higher correlation with the market price of risk when using ECB forecasts. The comparison between ECB 1 year and ECB current year forecast is made to determine if indeed the investors use a one year ahead forecast or are more short-sighted than expected and use a shorter term growth. Indeed, the 1 year ahead proves to be a better fit for the model, with higher statistical significance and a significantly higher R 2. The next comparison is between the different independent variables. The difference between the P t /P t-5 and P t /P t-5,6 was explained above. The averaged capital return P t /P t-5,6 performs marginally better across all specifications. For example, in the ECB 1 year ahead regressions the difference between P t /P t-5 and P t /P t-5,6 in R 2 is only 0.012, although it is higher for the Reuters Poll regressions. The coefficients themselves are almost identical. This is fortunate because it indicates that the regression results are not particularly vulnerable to the specific price in the time point selected in the past. Additionally, one of the goals of this thesis is to construct simple, easy to use models of expected returns. The fact that the resulting model can be used with the simpler P t / P t-5 without losing significant explanatory power makes the model easier to use Results - Regressions, Quadratic Model 1 The results that appear below are for the regression where χ t is alternatively P t /P t-5 and P t /P t-5,6 15

17 Model 1 Quadratic OLS Regression results ECB 1 Year Ahead Reuters 1 Year Ahead χt=pt/pt-5 χt=pt/pt-5,6 χt=pt/pt-5 χt=pt/pt-5,6 λ t-stat (-8.24) (-8.73) (-5.60) (-6.61) λ t-stat (5.80) (6.49) (4.49) (5.47) λ t-stat (15.78) (15.56) (10.08) (10.80) R Obser T- statistics are in parenthesis. The numbers of observations appear at the last row. Including χ 2 t in the regression tests for a varying marginal effect of observed wealth. Since λ 1 is still negative, while λ 2 (the interceptor of the squared price change) is positive, it indicates a decreasing marginal effect of wealth. As long as investors observe a positive price change in the index, they are willing to undertake more risk per unit of return; however this effect diminishes as the observed return grows larger. This could possibly indicate that investors are wary of very high past capital returns; they take into account that these returns are unlikely to persist into the future. 16

18 Once again, the ECB forecast has higher explanatory power, compared to the Reuters Poll forecast. Because the price change (or observed past returns in other words) has a higher predictive power when used with the ECB 1 year forecast, for both linear and quadratic versions, the λ time series derived from that growth rate is determined as the best fit for the models. To keep the results for the next models concise and easily comparable, only the ECB 1 year ahead forecast will be used. For the same purpose of conciseness, only the P t /P t-5,6 will be used in the next models, since it is a slightly better fit for the model than P t /P t-5. Comparing the linear and quadratic models, it is noticeable that the R 2 increases in the quadratic version. For P t-5,6 the R 2 is in the quadratic version against in the linear version. There is a drop of statistical significance, but still all the coefficients are significant at the 99% confidence interval Methodology - Two-Growth Rate Model (Model 2) The model above is designed for a short term expected return. The investment horizon is expected to be close to 1 year. The two growth model extends this horizon to a longer one. The longest horizon forecast available from the ECB staff is 5 years; therefore the model is configured with that horizon in mind. The difference between the short term model and this one is that growth rate. In the short term and the long term expected return model the current dividend is expected to grow in the 1 year ahead growth rate, such as:. However, in this long term model next year s dividend D 1 is expected to grow at the 5 year ahead growth rate since now the holding period is about 5 years. Therefore the dividend growth model gives the following: which can be rearranged to: (1b) where: g 1t = One-year ahead dividend growth rate as of date t, equaled to the GDP growth forecast g 5t = 5 year ahead dividend growth rate as of date t, equaled to the 5 year ahead GDP growth forecast from ECB. 17

19 (1b) is once again combined with the security market line: (2b) By equating (1b) with (2b) I end up with which is solved for the long term Sharpe ratio measure: (3b) Results Regressions for Model 2 The hypothesis for this model is identical to the short term model. The only difference is in the dependent value; λ t is now affected by the 5 year ahead growth rate: { Other forecasts besides the ECB 1 year ahead are not used and χ t only takes the values of P t /P t-5,6. The results of the regressions appear on the table below: 18

20 Model 2 OLS Regressions Linear Quadratic λ t-stat (-15.97) (-7.90) λ t-stat (5.28) λ t-stat (29.41) (15.28) R Obser T- statistics are in parenthesis. The numbers of observations appear at the last row All coefficients are statistically significant at the 99%, with especially high t-statistics in the linear form of the model. Additionally the R 2 is relatively high, reaching for the quadratic model. This indicates there is a high correlation between the return per unit of risk investors are willing to accept in the future and the observed past capital returns on the index. Comparing Model 1 and 2, the R 2 is slightly higher in the longer term Model 2. Since the independent variables are exactly the same, one can reach the following conclusion: Since Model 2 has a 5 year investment horizon it is sensible that the investors pay particular attention to the performance of the index 5 to 6 years ago to gauge future performance. The future horizon of Model 2 is a better fit with the past horizon of Model 2. The coefficients of Model 2 are also slightly higher in magnitude; although since they are of opposite signs the effects of the higher 19

21 constant cancels somewhat the effect of the higher negative observed price return variable. This is linked to the fact that the dependent variable is now higher in magnitude as well. The average Sharpe ratio of Model 1, using ECB forecasts, is while the average of the time series of Sharpe ratio of Model 2 is This is due to the difference of the 1 year ahead GDP growth forecast and the 5 year ahead (with the 5 year ahead being higher) mainly during the years Methodology - Two-Growth Rate, Term Structure Adjusted Model (Model 3) In order to produce a more fully fledged long-term expected return model, the slope of the term structure of interest rates is taken into account. It is presumed that investors are aware of it and may well take it into consideration when determining equity required expected rates of return of a long holding period. Thus, I include into the model the combination of short- and long-term rates which investors consider which is represented by the following: r 1t + β (r 5t -r 1t ) r 5t is the long term risk free rate, in this case of 5 years duration to coincide with the 5 year ahead forecast. β is the optimal combination of risk free rates according to the investors and is determined by the regressions below. For example if β is equal to 1, then r 1t + β (r 5t -r 1t ) = r 1t + r 5t - r 1t = r 5t in which case the target maturity in risk free rates is 5 years. This term adjustment is then subtracted from both sides of equation (1b) to produce: (1c) I divide by VIX resulting in having the Sharpe ratio λ t like previous models, only now it includes the term structure adjustment. That in turn I equate to the usual observed wealth accumulation equation: λ t = (for the linear version). To find out the unknown β I transpose the term structure adjustment to the right hand side of the equation: (2c) This specification will be used for the regressions below. 20

22 3.3.2 Results Regressions for Model 3 The specifications for the regressions are the following: { where Term t is the term structure adjustment in 2c. The results appear in the table: Model 3 Linear Quadratic λ t-stat (-15.09) (-7.45) λ t-stat (4.94) β t-stat (5.66) (5.33) λ t-stat (19.42) (12.98) R Obser T- statistics are in parenthesis. The numbers of observations appear at the last row 21

23 All coefficients are statistically significant at the 99% confidence interval. The R 2 is higher than previous models when comparing across the linear form and the quadratic form. The R 2 of the quadratic version now reaches 70% which points to a significant predictive power of the model. Since the coefficient of the Term Structure Adjustment is statistically significant and its inclusion improves the R 2 of the model, it could possibly indicate that indeed investors are aware of the slope of interest rates and take it into account when forming expectations of returns. Model 3 concludes the models designed to capture the relation of past observed wealth (returns) with the expected returns predicted by models using forward looking measures. It is noticed that overall the quadratic form models perform better (meaning they have a higher R 2 ) than the linear forms, indicating that there is possibly a decreasing marginal effect of observed past returns. Another conclusion is that the longer term models perform better than the short term models. Generally as the complexity of the models increase so does the R 2 as well. It is important to note however that even the simplest short term, linear model still has significant predictive power. Therefore the goal to obtain a simple model is accomplished. Moving away from the simple model, the higher complexity of the last model is counter balanced with a higher predictive power. The coefficients of these models are then used to form new models (see next chapter). This models now require only values for the risk free, the current price of the index divided with past price of the index at a certain interval ago (P t /P t-t ), and the VSTOXX index value. Additionally, the results of the model derived expected returns are compared with realized returns to test the correlation between the two and test if there is an equity premium puzzle with European data. 22

24 4. Data The focus of this paper is the performance of the FTSE Eurofirst 300 Index. A single benchmark equity asset is used in most models of equity returns as a proxy for market-wide equity performance. Since the focus of this thesis is European equity performance, I use the FTSE Eurofirst 300 Index. It is a capitalization-weighted price index which measures the performance of Europe's largest 300 companies by market capitalization and covers 70% of Europe's market cap. All data used throughout the thesis are annual measures, meaning the expected return is annual, the growth rate is the year on year rate and the volatility measure is annualized and so on. The frequency of the data is monthly. The main motivation for this is that all models are sensitive on the growth rate measure, which has a quarterly frequency. Using monthly frequency means that g 1t remains static for 4 observations each time. Using a daily frequency instead, for example, means that the g 1t would remain static for about 100 observations. To give a reference for the data described below, the derivation for the short term forward looking Sharpe Ratio is the following: The logic behind this relationship is explained in great detail under Methodology in Chapter 4. First, data concerning the equity asset were downloaded. The source of this data is the Reuters Datastream service. The base date of the index is December 31, The ticker of the Index is FTSEUR1ST 300 E. The values of the price of the index as well as the dividends are nominated in Euros. Therefore P 0 is simply the price of this index for every point in the time period. To give an overview of the price change of the index the following graph presents its variation during the time period: 23

25 1/1/1999 7/1/1999 1/1/2000 7/1/2000 1/1/2001 7/1/2001 1/1/2002 7/1/2002 1/1/2003 7/1/2003 1/1/2004 7/1/2004 1/1/2005 7/1/2005 1/1/2006 7/1/2006 1/1/2007 7/1/2007 1/1/2008 7/1/2008 1/1/2009 7/1/2009 1/1/2010 7/1/2010 1/1/2011 7/1/2011 1/1/2012 7/1/2012 1/1/ FTSEUR1ST 300 E - PRICE INDEX Performance of the Equity Index during the examined time period Besides the prices of the index, the dividend yield is also available and used directly on the models ( is essentially the dividend yield). The average for the dividend yield over the time period is %. The dividend yield as appears on the models is based on the annual dividend yield of the index. Dividends are also easily extrapolated. Since the dividend yield and the price is known, =x t can be solved for D 0t (where x is the value of the dividend yield for each t). Therefore the current dividend is a known variable in the models. By current dividend I refer to the sum of dividends paid over the course of the past year, up to the point t when the dividend yield is calculated. is the one year ahead dividend and it is an unknown variable from the point of view of time t, when constructing the expected return models. Therefore the following relationship is used, which is commonly found in valuation models. The current dividend D 0t is assumed to grow at a growth rate g 1t. Put in equation form: D 1t =D 0t (1+g 1t ). On the other hand, when dividends are used to calculate realized returns, dividends at the time t+1 are considered known variables. For example, to calculate the realized return of 2008 the price of the index in 2009 as well as the dividend paid until 2009 are needed. The point of view of realized returns is ex-post. Therefore the 2009 dividend paid is a known value. 24

26 To proxy for the growth rate of dividends g 1t in the expected returns models, a growth rate that is as broad as the European equity index above is needed. Therefore, I use the GDP growth rate for the Europe area. It is reasonable to assume that a European wide dividend should grow at the rate of overall European economy. This assumption is in line with my source of inspiration. DRG use the GDP growth rate of the US economy for the growth rate of the dividends of S&P 500. This growth rate needs to be a forecast from the point of view of each time t. In other words they are not realized growth rates. I use a time-series of GDP growth rates forecasts. In order to have a less biased model, I obtain data from 2 different sources. The first source is the European Central Bank (ECB) forecast, produced by specialized ECB staff and is widely accepted as a significant macroeconomic predictor. The second source is the Thomson Reuters poll, which captures the consensus of analysts and academics. The growth rate is a year on year estimate (annual). It is nominated in percentile terms. Both sources provide the forecasts on a quarterly frequency. To adapt this frequency to the monthly frequency of the rest of the data, I set each month s growth rate within each quarter equal to the quarter s growth rate. Therefore each month in the same quarter has the same growth rate estimate. It is important to note that it is the frequency that is quarterly. The growth rate is annual, in other words how much the European GDP will have grown after a year. I use 4 variations of the forecasts: -The one year ahead ECB forecast. The average forecasted year-on-year growth rate is 1.543% -The one year ahead Thomson Reuters Poll estimate. Its average is 1.321% -The current year ECB forecast. This mostly served as a test. If the current year forecast had more explanatory power than the one year ahead forecast, it could mean that the assumption that the dividend grows at the rate of 1 year ahead growth is erroneous. -The ECB 5 year ahead forecast. This is the longest horizon forecast available from ECB. It is used in the longer term models, namely Model 2 and Model 3. It is overall higher than the 1 year ahead forecasts. Its average is 2.178%. 25

27 The growth rate forecasts described above is the first forward looking measure. The second one is the implied volatility derived from the options market. The thesis, as well as my source of inspiration, does not use the Black-Sholes option pricing formula, or any formula for that matter to extrapolate the implied volatility. Instead it uses a volatility index that estimates this volatility by using option pricing formulas on put and call options on an equity index. Doran, Ronn and Goldberg use the VIX volatility index that measures the implied volatility of the S&P 500 index. The corresponding volatility index for Europe is the VSTOXX index. The VSTOXX Index is based on EURO STOXX 50 realtime options prices and is designed to reflect the market expectations of near-term up to long-term volatility by measuring the square root of the implied variance across all options of a given time to expiration. The reader will notice that the index Euro Stoxx 50 is different than the index I am using in the thesis. This is a limitation of my thesis. The reasons I do not use the Euro Stoxx 50 index as the equity asset proxy, although it is used in the volatility index, are the following: The Euro Stoxx 50 index performs very poorly during the time period. It has negative realized returns for extensive periods of time and it never reaches its starting price value again. Depending on the method I used to calculate the average realized return, it ranged from a meager 2% to a -1%. Such returns are too low when compared to previous financial literature on European equity returns. Also, the movement of the price is not very representative of the events of the time period. The FTSEUR1ST 300 Index, as can be seen by the graph above, responds consistently with significant economic events. For example, it recovers after the dot.com bubble and then subsequently drops sharply after the 2008 financial crisis takes effect. In contrast, the EUROSTOXX 50 index behaves randomly. Possibly, the reason for this is that the EUROSTOXX 50 index is narrower and thus less diversified than the FTSEUR1ST 300 Index. This is the second argument for using the FTSE index. Since the models are about a European-wide market premium, the broader the index is the better. The FTSE index is comprised of 300 equities of firms as opposed to only 50 of the EUROSTOXX index. It is therefore better in tracking the broader European economy. The risk free rate r 1t used for the short term is the yield of German government 6 month T-bills. It is the closest duration to the 1 year investment horizon of the model from available German 26

28 1/1/1999 7/1/1999 1/1/2000 7/1/2000 1/1/2001 7/1/2001 1/1/2002 7/1/2002 1/1/2003 7/1/2003 1/1/2004 7/1/2004 1/1/2005 7/1/2005 1/1/2006 7/1/2006 1/1/2007 7/1/2007 1/1/2008 7/1/2008 1/1/2009 7/1/2009 1/1/2010 7/1/2010 1/1/2011 7/1/2011 1/1/2012 7/1/2012 1/1/2013 government T-bills. I consider a risk free investment buying a short term fixed income security with extremely low probabilities of default and holding it until maturity. I use the yield because of the holding assumption. Modeling the risk free rate with the price change of the T-bill (instead of the yield) implicitly assumes risk of capital loses which escapes the definition of a risk free investment. Additionally, a long term risk free rate r 5t is needed for Model 3. The investment horizon of that model is 5 years, so a fixed income security of 5 years duration is needed as a proxy for that risk free rate. To keep the risk free rates consistent, German government bond yields are used again; this time for bonds with 5 year maturity. As an example the yield on German government 6-month treasuries is 3.01% on February of Therefore the risk free rate is equal to 3.01% for the models at that time t. It is interesting to notice the variation of the yields of treasuries. They range from 2% to 5% in the starting period but drop abruptly after the financial crisis of 2008, with the short term risk free rate reaching even negative values briefly. The risk free rates are plotted in the graph below. The Y axis is in percentages Risk Free Rates Long Term Short Term Performance of short term and long term risk free rates during the time period 27