CLAREMONT McKENNA COLLEGE

Transcription

1 CLAREMONT McKENNA COLLEGE INTERNATIONAL DIVERSIFICATION: AN OUT-OF- SAMPLE STUDY OF PORTFOLIO OPTIMIZATION USING ANALYSTS EARNINGS FORECASTS SUBMITTED TO PROFESSOR ERIC HUGHSON AND DEAN GREGORY HESS BY VALAY SHAH FOR SENIOR THESIS FALL 2007 DECEMBER 3, 2007

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3 Acknowledgements I would like to take this opportunity to thank everyone who has contributed in some form or another to this thesis. First of all, I would like to thank my family, especially my parents. Without their guidance, support, and encouragement, I would not be where I am today. I would like to acknowledge Professor Jennifer Ward-Batts and my fellow students in Econ 180 for their contributions, both content and style related. Finally, I would like to thank my reader, Professor Eric Hughson. His direction, suggestions, and insightful comments guided my dive into this challenge. He was always available to discuss issues and responded to s with astonishing speed. This thesis would not be what it is without his support. i

4 Abstract The equity premium, a stock s expected return in excess of the risk free rate, is used throughout finance. The extraordinary gains experienced by the U.S. stock market have resulted in a so-called equity premium puzzle, leading many academics to expect much lower returns in the future. This paper examines the out-of-sample performance of internationally diversified mean-variance optimized portfolios that use ex-ante equity premium forecasts and compares it to the out-of-sample performance of portfolios derived from historical data. As was expected, the portfolios created using the equity premium estimates significantly outperformed those portfolios constructed using historical data. Somewhat more surprising was that these portfolios also showed consistently higher Sharpe ratios than portfolios constructed from heuristics. This indicates that analysts earnings forecasts may have some predictive power. ii

5 Table of Contents I. Introduction..1 II. Literature Review 7 III. Theoretical Framework..14 IV. Data 18 V. Methodology..27 VI. Results 30 VII. Discussion of Results 33 VIII. Conclusion 36 IX. Figures and Tables 38 X. References.48 iii

6 List of Figures and Tables Figure 1: Efficient Frontier 38 Figure 2: Efficient Frontier 39 Table 1: International Correlation Coefficients...39 Table 2: Model Based Expected Rates of Return 40 Table 3: Historical Means and Standard Deviations 41 Table 4: Optimal Portfolios.42 Table 5: Historical Index Returns for Out-of-Sample Period..45 Table 6: Out-of-Sample Returns..46 Table 7: Summary of Sharpe Ratios 47 iv

7 1 I. Introduction Portfolio diversification is of vital interest to investors. In today s global and integrated economy, investors are not resigned to diversification solely within their domestic market. Rather, they are able to span the globe for assets which can diversify their portfolio and improve their overall risk-adjusted performance. As a result, the proper estimation of the ex-ante equity risk premium from various markets is of vital importance to an investor seeking to minimize their risk-to-reward ratio. Optimal asset allocation is one of the most important topics in modern financial theory. The equity premium, a stock s expected return in excess of the risk free rate, is used throughout finance. It is a required input in the calculation of the cost of capital used for capital budgeting decisions, used in long-term forecasts of investment portfolio growth, and critical in determining an expected rate of return on stocks (Goetzmann & Ibbotson, 2005). However, attempts at quantifying and calculating the equity premium primarily came about as a result of the Capital Asset Pricing Model (CAPM), which requires the equity premium as a critical input (Goetzmann et al., 2005). The importance of forecasting the equity premium is clear. Accurate forecasting of expected stock returns would allow for higher reward to risk ratios when adopting a mean-variance framework for portfolio optimization. Mean-variance analysis requires as inputs the expected return and variance in return for each asset in the portfolio, as well as a correlation matrix that details the correlations between the returns of each asset. Meanvariance analysis often uses historical data as inputs for future expected returns and

8 2 expected variances of each asset. The result of the mean-variance process is the selection of an ex-ante so-called optimal portfolio, which maximizes expected return for a given level of risk. However, when taking estimation error into account, this approach no longer guarantees an optimal portfolio selection, and conversely, research has shown that the experienced returns under these conditions are inferior to adjusted approaches (Campbell & Thompson, 2005; Smith, Steinberg, & Wertheimer, 2006). Michaud (1989) argues that due to input errors and lack of information, some naïve heuristic strategies would outperform optimized portfolios. Nearly all of the research in the field in the past agrees with the fact that the use of historical data as inputs will produce inferior optimal portfolios to other methods, including the use of consumption based models and surveys. Alternatively, the required inputs can come from models. If an investor were attempting to optimize a portfolio of stocks, he or she could use a model such as the Capital Asset Pricing Model (CAPM). However, even if you were using such a model, an investor would require knowledge of the equity premium. In the case of international diversification, the assets which comprise the optimal portfolio are indices of varying countries. It turns out that the equity risk premium is extremely difficult to estimate for future periods. The equity premium puzzle, as it was first coined by Mehra and Prescott (1985), refers not only to the significant amount by which stocks have outperformed bonds in the past century, but also to the fact that investors would have to be extremely risk-averse in order to explain the historical equity premium. They claim that investors would be required to have coefficients of relative risk aversion of 30 or more in order to account for the high level of the historical equity premium. In numerical terms, this

9 3 means that an investor would be indifferent between a certain payout of $51,209 and a bet that offers a 50% chance to win $50,000 and a 50% chance to win $100,000 (which has an expected payout of $75,000) (Mankiw & Zeldes, 1991). Mankiw and Zeldes argue that this is an unreasonable level of risk-averseness to expect from nearly any investor. Mehra and Prescott believed that theory only supported equity premiums up to a level of 1% at most. This was significantly lower than their estimated historical annual equity premium of 6.2%. Dimson, Marsh, and Staunton (DMS) (2006), thus, characterize the equity premium puzzle as a quantitative puzzle about the magnitude, rather than the sign, of the risk premium (page 1). They offer two potential resolutions. The first states that the model used by Mehra and Prescott is wrong. Since their paper in 1985, many researchers have modified the model through the amendments of their core assumptions, as well as the inclusion of several new assumptions. DMS note that although some of these adapted models may solve the puzzle, they require significant changes to the original model and still require the investor to have high levels of risk aversion. The second possible resolution to the equity premium puzzle that they offer is that the historical premium is deceptive, and investors should lower their expectations for future equity risk premiums. They postulate that the extraordinarily high equity premium may be the result of country selection bias. They note that when there is ex post selection bias based on past success, historical mean returns will provide an upward biased estimate of future expected returns (page 2). The majority of the research has focused on the use of in-sample regressions where the fitted values were compared to the actual values. However, as Goyal and

10 4 Welch (2004) point out, a real-world investor would not have had access to any ex-post information... An investor would have had to estimate the prediction equation only with data available strictly before or at the prediction point, and then make an out-of-sample prediction (page 2). Truly powerful diversification methods ought to outperform heuristic approaches to optimal portfolio selection in out-of-sample testing on a risk-toreward basis. This paper examines the out-of-sample performance of internationally diversified mean-variance optimized portfolios that use ex-ante equity premium forecasts and compares it to the out-of-sample performance of portfolios derived from historical data. Using Claus and Thomas s (CT) (2001) ex-ante estimates for the equity premium for 4 international markets, mean-variance optimized portfolios were constructed for each year of available data. The four countries are Japan, the United States, the United Kingdom, and Germany. CT derive two equity premiums, one using an abnormal earnings model and one using a constant growth dividend discount model. Optimal portfolios were constructed using both sets of data as inputs. Simultaneously, optimal portfolios were constructed using historical data as inputs. The performances of these three portfolios were then measured out-of-sample and compared with the out-of-sample performance of two benchmark portfolios: the minimum variance portfolio and the equally weighted portfolio. Optimal portfolios and out-of-sample performance were calculated for 1, 3, 5, and 10 year holding periods to determine the impact of investment horizon. Sharpe ratios were calculated to determine which diversification method provides the highest rewardto-risk ratio.

11 5 This paper has two main objectives. The first is to determine whether or not the incorporation of expected equity premiums far below the prevailing historical averages is advantageous, in the form of superior risk-adjusted returns, to an investor utilizing a mean-variance framework. The second is to determine whether or not there is truly any predictive power in analysts earnings forecasts, which are the main drivers behind the ex-ante equity premium estimates of CT. A consequence of the previous result would be to determine which model, between the abnormal earnings model and the constant dividend growth model, is a better estimator for ex-ante equity premiums in international markets. It did not come as a surprise that the portfolios constructed using sample averages for expected mean were significantly outperformed by each of the other four methods. In fact, the experienced returns of these portfolios had a negative Sharpe ratio for every holding period. Contrary to Michaud s (1989) work, the Sharpe ratios of the out-ofsample returns for the abnormal earnings model based portfolios and the constant dividend growth rate models were significantly higher than the Sharpe ratios for the two benchmark portfolios. The next section reviews some of the existent literature pertaining to international diversification and equity premium estimation. Due to the extensive amount of research that has been undertaken with regards to the equity premium puzzle, the literature review is not meant to be an exhaustive painting of the landscape of previous work, but rather representative of the general theories and views held on the specific concepts. Following the literature review is an examination of the theoretical underpinnings of mean-variance analysis. A comprehensive look at how CT arrived at their ex-ante equity premiums

12 6 comes next, followed by an overview of the methodology of portfolio construction and performance comparison used in this paper. The results and conclusion bring the paper to a close.

13 7 II. Literature Review There is an exorbitant amount of literature that discusses why using historical return as an ex-ante estimate is a poor estimator in the mean-variance framework (Arnott & Bernstein, 2002; Dimson, Marsh, & Staunton, 2006; Jorion, 1986; Yamaguchi, 2005). Jorion notes the significant amount of estimation error that arises from the use of sample mean as expected mean. Fischer Black states explaining average return is like explaining variance, but does little to help us estimate expected return (as cited in Yamaguchi, page 2). There has also been significant research which postulates that the historical mean for the equity premium does not apply to more recent periods, and that the expected future equity premium is much lower (Arnott & Bernstein, 2002; Bernstein, 1997; Claus & Thomas, 2001; Donaldson, Kamstra, & Kramer, 2007). Arnott and Bernstein argue that the investment management industry thrives on the use of forecasting the future by extrapolating the past. They believe that U.S. investors have come to expect stocks to produce a 5 percent risk premium over bonds. They believe that these expectations are rooted in the collective psyche of the investment community (page 2). The majority of the research regarding the equity premium has focused on showing different variable as significant estimators of the equity premium through the use of in-sample regressions (Longstaff & Piazzesi, 2004; Lucas, 1978; Mehra and Prescott, 1985). Goyal and Welch (2004) consider the dividend yield, interest and

14 8 inflation rates, the book-to-market ratio, and the earnings-price ratio, among others, as variables with in-sample predictive ability. They find that not one of these variables had significant out-of-sample predictive ability for an investor with access to only ex-ante information. Additionally, others have used consumption based models to accurately predict, ex-post, the in-sample equity premium. In general, researchers come up with a consumption based model, and then adjust the model and their assumptions so that the model becomes a good in-sample predictor for the historical equity premium. That is to say, rather than determining what the true value of the equity premium ought to be based upon theory, they mold their models based on the assumption that what has been observed in the path is the true behavior of the equity premium. Much of the initial research surrounding the equity premium made use of consumption based models. This is due to the fact that Mehra and Prescott (1985) themselves used a consumption based model. Theirs was a variation of Lucas s (1978) single asset exchange model. They find that the equity premium puzzle may not be about justifying the high returns, on average, witnessed in the stock market, but rather about why the risk-free rate returns were low. Longstaff and Piazzesi (2004) expand on Mehra and Prescott s work with the inclusion of corporate cash flows in their model. These cash flows represent a minute portion of aggregate consumption, but are extremely volatile. They conclude that the equity premium is wholly determined as the sum of three separate risk premia: corporate risk, event risk, and consumption risk. They then adjust this model to reflect historical data. Longstaff and Piazzesi conclude that their model suggests that the equity premium is 2.26%, much higher than first postulated by Mehra and Prescott, but still too low to have solved the equity premium puzzle.

15 9 Not all of the model based research regarding the equity premium revolves around consumption based models. Akdeniz and Dechert (2007) note that Brock s asset pricing model, which is a production based asset pricing model, allows for much higher equity premiums than Lucas s consumption based model. However, there is no closed form solution to the equation and they can not come to a specific conclusion about whether or not the equity premium puzzle is invalid under their assumptions. Fama and French (2002) also estimate the equity premium using both an earnings model and a dividend discount model. The main difference in their model is that they assume that both the compounded dividend growth rate and the compounded earnings growth rate must approach the compounded capital gains growth rate, assuming a large enough sample size and stationarity in the dividend-price and earnings-price ratios. Their estimated equity premiums for the period from are 2.55% and 4.32% for their two models, compared to the actual historical value of 7.43%. They claim that their estimates are closer to the true value of the equity premium than the historical data, and that the exceedingly high stock returns for the period can be attributed to lower expected returns. All of these studies share something in common. They assume that the equity premium is constant over time, due to the efficient market hypothesis. This is also used as justification for associating the expected rate of return to that discount rate which equates the dividend or earnings growth models to the prevailing market valuation of the security. Campbell (2007) estimates the equity premium while allowing for changes in the theoretical value over time. He also accounts for uncertainty by incorporating a random walk for the dividend-price ratio into the Gordon growth model. Campbell s estimate for

16 10 the current equity premium for the United States rests at just over 3%. However, he restricts himself to predictability in short-term regressions and includes no discussion on the uncertain nature of predicting equity premiums. Harris and Marston (2001) equate IBES earnings forecasts to future cash flows using the Gordon growth model, similar to CT. They find a mean ex-ante equity premium of 7.33% in the U.S. for the period from Although CT downplay the inferences that can be made using the Gordon growth model, Harris and Marston argue that their results suggest an equity premium that is close to the historical spread between bond and stock returns. Rather than interpreting this as an absolute figure, they assert that their values establish an upper bound for the equity premium. Their study also focuses on determining the impact of various ex-ante measures of risk, such as interest rates, consumer confidence, the distribution of earnings growth forecasts, and implied volatility on the S&P 500 index through the use of options data. They come to the conclusion that the equity premium is not a constant value. Rather, it changes over time and is affected by these various measures of risk. Donaldson, Kamstra, and Kramer (2007) also agree with Campbell, in that they believe equity premium models that allow for time-variation, breaks, and/or trends are critical features of the equity premium process (page 1). By observing various fundamental statistics over various eras, such as dividend and earnings yields, they are able to use their model, which yields ex-post estimates of the equity premium, in conjunction with current fundamental statistics in order to make an ex-ante estimate of the current equity premium. They come to the conclusion that this value is extremely close to 3.5%. Dimson, Marsh, and Staunton (2006) remark that the equity premium

17 11 could have been non-stationary due to a number of factors, including changes in the opportunities available to investors to diversify and the attitude of investors towards risk. Past reductions in the equity premium would have resulted in lower discount rates, which would have subsequently led to higher stock valuations. As a result, they argue, past reductions in the equity premium inflate historical returns. At the same time, there has not been much research concerning the equity premium puzzle in international markets. Canova and De Nicolo (2003) observe that equity premium puzzle has been studied almost solely for the United States, with the major exception being Campbell (1999), while international evidence has for the most part been ignored. CT use analysts forecasts to predict the ex-ante equity premium for several international markets. They find that the historical equity premium is too high for more recent periods and that 3 percent is a likely upper bound for the equity premium for all of the markets observed heading into the future. The U.S. had the highest ex-ante average equity premium at 3.4% from Shackman (2006) studies the equity premium puzzle from an international perspective. Calculating returns above the risk free rate for 39 countries, Shackman discovers that emerging markets experience much higher excess returns as compared to developed markets. However, due to the large volatility of these markets, the riskadjusted returns of developed markets tend to be higher. Testing for potential reasons developed markets experience higher risk-adjusted returns, Shackman finds that there is a positive correlation between returns above the risk-free rate and the degree of integration of domestic markets with global markets.

18 12 Others (Hail & Leuz, 2004a and 2004b; Mishra & O Brien, 2005) have reviewed the empirical relationship between various estimated risk factors, which include stock return volatility and political risk, and ex-ante expected returns for emerging markets. Mishra and O Brien use IBES earnings data for individual firms from emerging market countries, along with CT s form of the residual income model, in order to estimate the ex-ante equity premium for these markets. In order to determine the determinants of expected return for these markets, they regress the ex-ante risk premiums on three measures of risk: the local beta, the global beta, and a variable called political risk, which is defined as the ratio of return volatility of the stock to the return volatility of the global market (page 11). Their result, consistent with previous literature, shows that the single most significant risk factor is political risk. Additionally, global beta increases the model s explanatory power only if that specific market provides substantial investability to global investors (page 13). There have been multiple approaches to adjusting mean-variance analysis to account for this lack of real-world practicality from using historical data. Much of the literature revolves around deriving ex-ante estimates of the equity premium as mentioned earlier. However, a recent focus of study has been to examine the effectiveness of using survey data as inputs. Smith, Steinberg, and Wertheimer (2006), who used the Livingston Survey, argue that the use of predictive, or forward-looking, estimates in portfolios not only outperforms portfolios constructed with historical data, but it has the added attribute that it allows one to take into account one s own beliefs. Of course, not all of the literature agrees with Smith, Steinberg, and Wertheimer. Söderlind (2007) argues that out-of-sample forecasting of traditional models tends to provide better results than

19 13 forecasting using the Livingston Survey. He claims that the use of surveys results in analogous outcomes to that of a too large forecasting model: poor performance and too sensitive to irrelevant information (page 1). The majority of the literature in the subject matter of the effectiveness of using surveys in stock market forecasting, at least pertaining to the US equities market, has been limited to the use of the Livingston Survey (Dokko & Edelstein, 1989; Lakonishok, 1980; Pearce, 1984; Söderlind, 2007). There has been research done (Vissing-Jorgenson, 2003) on the behavioral finance as it relates to expectations of investors. However, the author did not actually test investor expectations in a simulated model to determine portfolio performance. In that particular study, the author used the UBS/Gallup poll conducted in 1996 and concluded that there was evidence of biased self-attribution (page 143). This occurs when investors who have observed strong performances in the recent past attribute it to their level of skill, while those investors who have not fared as well blame it on poor luck. As a result, investors who have performed well in the past have expectations for continued strong performances in the future, while those who have performed poorly feel as if they will perform better in the future. Biased self-attribution may be a source of estimation error in analysts earnings forecasts as well.

20 14 III. Theoretical Framework Mean-variance analysis is the study of portfolio theory in which multiple individual assets are combined to create a portfolio. It is called mean-variance analysis because there are only two significant variables: the mean (return) of the asset and the variance of the asset. The process allows the investor to derive that portfolio that, for a specific risk tolerance level, will yield the highest rate of expected return. This portfolio is a weighted average of each of the individual assets considered. The equations below detail the calculations for the portfolio s expected return and portfolio s expected variance given the expected return and expected variance of each asset to be included, as well as the correlation matrix. (1) ( rp ) =! E w * E( r ) i i i Here, r p is the expected return of the portfolio, w i is the percentage of the portfolio that is made up of the i th individual asset, r i is the expected return of the i th individual asset, and! (2) w = 1. i i 2 (3) " ( ) =!! wi w j" ij =!! p w w " "! i j Here,! 2 (p) is the expected portfolio variance, w i and w j are the weights of the i th and j th individual assets,! i and! j are the expected standard deviations of the i th and j th individual assets,! ij is the covariance between asset i and asset j, and " ij is the correlation coefficient between the i th and j th individual assets.! i j i j i j ij

21 15 The concept of mean-variance analysis was developed in the by Harry Markowitz (1952). The efficient frontier, which is the collection of all of the portfolios that yield the highest expected return for their respective levels of risk, is called the Markowitz Frontier. Markowitz s work revolutionized finance in that he was able to show that the addition of an asset which was riskier than the portfolio could lower the overall risk of the portfolio. Equation (3) details the formula for portfolio variance. Diversification reduces portfolio variance unless returns are perfectly positively correlated. However, portfolio variance will not go to zero because the correlation between asset returns is usually positive. Elton and Gruber (1973) estimated that the average correlation coefficient for U.S. equities was likely in the 35-38% range. Thus, it is clear that Markowitz s work is the key to modern day attempts at portfolio diversification. Mean-variance analysis uses expected return and expected variance in return of each asset, along with a matrix detailing the correlation coefficients of each asset. Using this data, and the equations detailed above, the set of all possible portfolios is created. Each member of the set is expressed as a combination of the expected return and expected variance of that specific portfolio, and can be graphically represented as a point on an x-y axis where the x-coordinate is the portfolio s expected standard deviation and the y-coordinate is the portfolio s expected return. Each point represents one possible group of weights for each asset in the portfolio. Once this set is known, the efficient frontier can be determined. The efficient frontier is the set of points that yield the highest expected return for each level of expected variance. All points below the frontier are discarded because there is another portfolio that can be created which yields a higher expected return for the same level of risk. Figure 1 shows two such portfolios. Portfolio

22 16 A and portfolio B have the same expected risk level; however, portfolio A has a higher expected return. No rational investor would choose portfolio B over portfolio A. Since portfolio A has the highest possible expected return among all potential portfolios with the same level of expected risk, it lies on the efficient frontier. Traditional mean-variance analysis uses historical data as inputs for future expected returns and expected variances of each asset. A time-series analysis is typically conducted on historical data, and the resulting estimates for returns and variances are used as accurate inputs without accounting for possible estimation error. At this point, the efficient frontier is determined. Tobin s Separation Theorem (Tobin, 1958) is used when there is an inclusion of a risk-free asset, typically some form of U.S. Treasury Bond with the appropriate time horizon. Tobin s argument was essentially that the optimization process could be broken down into two core steps. The first is to create the efficient Markowitz frontier as described above. The next step is to combine this frontier with a risk-free asset. The investor then chooses the appropriate amount of risk that he or she is willing to take on, and the optimal portfolio lies somewhere on the line connecting the risk free rate and this indexed asset. The tangent line is known as the Capital Market Line (CML). The Sharpe ratio, originally introduced in 1966 by William Sharpe as the reward-to-variability ratio, is a measure of the additional return a portfolio achieves per additional unit of risk it takes on (Sharpe, 1966, page 123). Each portfolio on the efficient frontier has the highest Sharpe ratio of all portfolios with the same level of expected risk, and the tangent portfolio is the portfolio with the highest overall Sharpe ratio.

23 17 There is one other significant indication of Tobin s Separation Theorem. It allows for the investor to borrow at the risk-free rate and leverage the portfolio in order to achieve higher levels of expected return, while also taking on higher expected risk. Figure 2 shows this concept. Points A, B, C, and D all represent various achievable portfolios. Portfolio A lies on the tangency between the risk-free rate and the tangent portfolio. This represents a new portfolio, which is a weighted average of the original portfolio of risky assets and a second asset, one that has an expected return equivalent to the risk free rate and a standard deviation of zero. Point B is the tangent portfolio. This portfolio represents a portfolio where you carry no T-Bills and one hundred percent of your money is put in the original portfolio of risky assets, with the money being split based on the original weights, w i. Point C lies past the tangent portfolio. This portfolio represents a leveraged portfolio. It essentially entails have a weight on the risk free asset of less than 0, and a weight on the risky asset portfolio of greater than one. It represents that portfolio that can be created by borrowing at the risk free rate in order to purchase more of the risky portfolio. In practice, an investor can not borrow infinite amounts at the risk free rate, and thus, the line would tend to tail off after a while as shown in the figure. Point D lies on the y-axis and represents that portfolio which consists solely of T-Bonds, the risk free rate.

24 18 IV. Data The goal of this study is to determine the effectiveness of using ex-ante equity premium estimations in an internationally diversified portfolio and compare its out-ofsample performance with that of a portfolio derived from historical data. The tangency portfolios are determined using ex-ante estimations of the mean of international indices as well as historical average returns. The risk free rate, given by the investment horizonappropriate Treasury rate, is taken from an online database of historical data from the U.S. Federal Reserve. Adjusted close figures have been collected for four major international indices: the S&P 500 (USA), the Nikkei 225 (Japan), the DAX (Germany), and the FTSE (United Kingdom). The historical data for the DAX is from the Deutsche Bundesbank. The historical data for the Nikkei 225 and the FTSE 100 indices come from Wren Research, who has collected the data from the Financial Times and the Bank of Japan. The historical data for the S&P 500 comes from Yahoo! Finance. The adjusted close value represents the value of the specific index at the end of the trading day adjusted for any distributions that were made. These distributions can include cash and stock dividends, as well as stock splits and rights offerings. The adjustments for splits and right offerings are necessary so that the close value accurately reflects the change in value from the previous day. If a stock were to split 2-to-1, the price of the individual stock would be reduced by fifty percent, however, the index value should not change because no value has been gained or lost. Dividends must be included because we wish to use a

25 19 measure of realized return. As equation (4) shows, realized return is defined as capital gains plus dividends received divided by the original purchase price. (4) P1! P0 + r s = P 0 D The data for the ex-ante estimated equity premiums come from CT s study. Theirs is one of the main studies focused on estimating specific ex-ante equity premiums for international markets. Additionally, their results include estimated premiums much lower than historical averages, and come from analysts earnings forecasts. Thus, by testing the out-of-sample performance of portfolios created from these equity premiums, it can be determined if there is any predictive power in analysts earnings forecasts, as well as if there is any benefit to inputting lower expected equity premiums. They collected data from the Institutional Brokers Estimate System (IBES). This database provides information regarding all analysts earnings forecasts for publicly traded stocks. They collected data at the same time each year, so that the risk-free rate would be the same across each annual sample. Since the fiscal year end is typically at the calendar year end in December, and the SEC requires financial statements to be filed within ninety days of this fiscal year end, they collect forecast data as of April for each year. CT claim that the estimated equity premium is that discount rate which equates the present value of expected future cash flows, observed via earnings forecasts, with the market valuation of that stock. They use two separate models in estimating the equity premium: the Gordon dividend growth model, and the abnormal earnings model. The dividend discount model

26 20 has been used in other papers as well. Harris and Marston (1999) also equate IBES earnings forecasts to future cash flows using the Gordon growth model; however, they undertake this examination for only the S&P 500. They find a mean ex-ante equity premium of 7.33% for the period from , compared with CT s mean equity premium of 7.34% over the same period using the same model. Since CT s paper includes the use of the Gordon dividend growth model for our international indices as well, this data is used to construct a third portfolio with which to compare out of sample performances. The dividend growth model of Gordon is a special case of the Williams (1938) dividend discount model detailed in equation (5) below. The Williams dividend discount model, detailed in his 1938 book The Theory of Investment Value, was the amongst the first studies that examined the concept of discounted cash flows as a valuation technique, but was the first to come up with the dividend discount model as a stock valuation method. It equates the value of a stock with all net future cash flows. Since dividends are all cash flows paid to the shareholder, the model values a stock as the net present value of all future dividend payments. In equation (5), k* is the discount rate. When equating the net present value of all future dividend payments to the current market valuation of the given security, k*, as the discount rate, is the expected rate of return on that security assuming efficient markets. (5)! di p 0 = = ( 1 + k i 1 * ) i The Gordon growth model, also known as the constant growth dividend discount model, assumes that each future dividend will grow at a constant rate, g, to calculate the

27 21 present value of a stock given a required rate of return. Equation (6) details the constant growth dividend discount model, which implies that the required rate of return, which in this case is the expected rate of return since we are equating current market valuation with future cash flows, is equivalent to this perpetuity growth rate g plus the one period forward dividend yield (d 1 /p o ). Since CT wish to deduce the equity premium based upon earnings forecasts, they solve for that discount rate which equates current market valuation of a stock with the model, using forecasts as inputs for future cash flows. d *(1 + g) ( 1 + k ) d k! g (6) p0 =! k* = + g * i * = " i= 1 i! 1 The problem with the constant growth dividend discount model is in determining what rate g is appropriate to assume dividends will grow at in perpetuity. Historically, researchers have tended to use the projected growth rate of earnings for the next 5 years, referred to as g 5. CT debate the positives and negatives of using this rate as the assumed perpetuity growth, noting that the majority of the research has focused on the United States domestic market. Additionally, there may be certain incentives for analysts to make optimistic forecasts in order to curry favor with management, but they might also adjust their estimates downward to be able to meet or beat them when announcing earnings (page 7). They come to the conclusion that g 5 is an optimistic assumption for infinite future dividend growth. Since they are trying to show that the historical equity premium is too high, having an optimistic growth rate would yield higher ex-ante equity premiums. If they can show that despite these optimistic assumptions, the ex-ante equity premiums generate still are short of the historical data, their argument would be d p 0

28 22 strengthened. As a result, CT incorporate g 5 as their value for the constant growth rate in the Gordon growth model. The abnormal earnings model, in their opinion, diminishes many of the difficulties that arise with the use of the constant growth dividend discount model. Equations (7), (8) and (9) detail the basis for the abnormal earnings model. Essentially, the abnormal earnings model states that the current price of the stock is equivalent to the current book value of equity plus expected future abnormal earnings discounted to present value. Expected abnormal earnings are forecasted future earnings adjusted down to reflect a cost of equity charge, as detailed by equation (9). Note the difference in notation between k and k* used by CT. k* represents the expected rate of return when equating the net present value of future dividends to the current market valuation of the stock using the Gordon growth model, whereas k represents the expected rate of return derived using the abnormal earnings model. (7) d e! bv! bv ) t = t ( t t! 1 (8)! " aei p 0 = bv0 + = ( 1 + k) i 1 e t = earnings forecast for year t bv t = expected book value of equity at year t end ae t = expected abnormal earnings for year t k = expected market rate of return (9) ae e! k bv ) t = t ( t! 1 i Equation (7) above is included in order for the model to be consistent with accounting theory. It is known as the clean surplus relation, [which] requires that all items affecting the book value of equity, (other than transactions with shareholders ) be

29 23 included in earnings (page 7). CT note that there are several transactions which, under United States accounting rules, would be able to circumvent the income statement, thus violating this relationship. In the end, however, it is of no significance in this study. These transactions occur ex-post, and thus are not reflected in analysts forecasts of earnings. Abnormal earnings are used as a proxy for economic profits, which are also known as rents. The IBES database that is used to collect the earnings forecasts only provides this data for the next five years. Thus, the authors assume that abnormal earnings grow at a constant rate, called g ae after year 5. As a result, equation (8) is adjusted to equation (10), which equates the market value of the stock with the current book value of equity plus the forecasted abnormal earnings for the next five years discounted to present value plus all future forecasted abnormal earnings, which are grown to perpetuity, at this constant rate g ae. In corporate finance terms, the bracketed value is known as the terminal value, and is the present value of all abnormal earnings past year 5. It is arrived at through an application of the constant growth dividend discount model. In this case, the next period dividend is equivalent to the forecasted abnormal earnings for year 6. This value is equivalent to the present value of abnormal earnings for year 5, (ae 5 )/(1+k) 5, grown at a rate g ae. This value is then grown at perpetuity by a rate of g ae. This rate is assumed by the authors and is equated to the expected inflation rate. The expected inflation rate is derived from the risk-free rate, with the additional assumption that the real risk-free rate is roughly three percent. CT consider other values for g ae and show that their estimated risk premium is relatively robust to variation in the assumed growth rate (page 13). As with their optimistic assumptions surrounding g, CT s base abnormal earnings growth

30 24 rate of r f 3% is larger than any growth rate that has been assumed in previous literature surrounding abnormal earnings. Thus, their low ex-ante estimated equity premiums using the abnormal earnings model are not a result of manipulated assumed perpetuity growth rates. 5 ae =! ( 1 + k) ae ( 1 + g ) ( k! g )(1 + k) i 5 ae (10) p0 bv0 + + [ ] i 5 i= 1 CT then make some final assumptions. They note that expected rates of return are most likely stochastic. This essentially means that the distribution of expected rates of return is most likely a random, or non-deterministic, process, in that no state fully determines the next state. However, they cite research that shows that k*, the estimated market discount rate using the dividend discount model, and k, the estimated market discount rate using the abnormal earnings model, are discount rates which are not stochastic. Except for a few recent studies, the majority of the research inherently assumes that the discount rates can approximate the expected rate of return. As a result, they also make this assumption. For the purpose of calculating an equity premium, the authors use the prevailing ten-year risk free rate for each specific country as of April of each given year. Thus, the estimated equity premium is simply this discount rate, or expected rate of return, minus this risk free rate. Additionally, although the equations above assume a constant required rate of return over the entire time period, these equations can be adjusted to incorporate changes in this rate over time. CT specifically consider the case when discount rates vary over future periods, based on the term structure of risk-free rates (page 9). However, the ex- ae

31 25 ante estimated equity premium is still assumed to be constant. As a result, equation (10) is adjusted to equation (11) below. p 0 = bv +! (11) t= 1! 0 " t s= 1 ae ( 1 + r t fs + rp) r fs = forward one-year risk-free rate for year s, rp = equity risk premium, assumed constant over all future years, ae t = expected abnormal earnings for year t, given by e t bv t-1 (r ft +rp) for the first five years following the forecast year, and ae 5 (1+g ae ) t-5 for each year thereafter into perpetuity. CT also provide a significant theoretical framework for why the abnormal earnings model in equation (11) is superior to that of the Gordon growth model detailed in equation (6). The key difference lies in the difference between how much of the socalled value profile is fixed by variables that do not need to be assumed. This value profile is defined as the fraction of total value captured by each future year (page 9). In the abnormal earnings model, the present book value of equity and the abnormal earnings forecasts for years one through five are not assumed by the researcher. However, in the constant growth dividend discount model, all of the value profile is determined specifically as a function of g, the assumed growth rate. In addition, there is much more reliability surrounding the reasonable ranges for g ae as opposed to the range for possible values of g. As noted earlier, the authors conduct a sensitivity analysis of their estimated risk premium subject to variations in g ae, the assumed abnormal earnings growth rate. They find that increasing the g ae by an additional three percent, which is equivalent to assuming a perpetual 3 percent real growth in rents and far larger than any assumed rate in previous literature, results in an average estimated ex-ante equity premium of 4.66%

32 26 over their 14 year period. This value, compared with an average of 3.40% for their original growth rate, is still far below historical estimates of the equity premium. Despite these advantages, both models share one significant flaw: neither takes into account uncertainty, especially that uncertainty which arises from estimation error on the part of analysts forecasts. Forecasted earnings may not be assumed by the researchers, however, analysts forecasted earnings are assumed to be true. As a result, the expected rates of return may suffer greatly from estimation error.

33 27 V. Methodology For this study, the portfolios that have been constructed include 4 risky assets, which are the 4 market indices noted earlier. The appropriate weighting of these assets were determined using historical data for returns as well as the two models for each year from The correlation coefficients between the risky assets are a required input in mean-variance analysis. Table 1, the correlation coefficients between the international indices, comes from the International Finance and Accounting Handbook (Choi, 2003). Although the coefficients are arrived at using data from , which would make these ex-post correlation coefficients, Choi notes that his results are similar to the findings of other authors using previous timeframes, and would thus be similar to reasonable ex-ante estimates of the correlation coefficients between international indices. Through the use of financial spreadsheets, the appropriate weightings for each of the four indices in combination with the risk free rate can be determined. The other required inputs are expected variance of returns, arrived at from historical data, and the expected returns, which come from historical data and the two models. Table 2 is the input data for expected return for the two models directly from the CT paper. They did not calculate estimates for each year for each country because not all countries had a sufficient number of firms with estimates in the database for each year. The historical means and variances were calculated going as far back as the dataset would allow. This was 1960 for the DAX, 1970 for the FTSE, 1970 for the Nikkei, and 1950 for the S&P 500. Table 3 details

34 28 the historical means and standard deviations for each index in the set for each year from 1991 through Since the estimated equity premiums are ex-ante values, the standard deviation in estimated equity premiums is not an accurate proxy for expected standard deviation. Additionally, since it would require the knowledge of future estimated equity premiums in order to calculate the expected standard deviation to use in the first year, this would be an ex-post expected volatility. This information would not be available to an investor making an asset allocation decision, and thus is not appropriate to use in this case. The historical volatility is more appropriate and is thus used in combination with the ex-ante estimated equity premium as inputs. These weightings were then used to create a sample portfolio in the past and what the actual one year performances would have been for each of the three optimal portfolios are compared given these specific portfolio weightings. That is to say, the optimal portfolio derived under each scenario (historical, dividend growth model, and abnormal earnings model) is taken, the actual purchase of that portfolio is simulated, and the return that would have been achieved had the investor held that portfolio for one year is calculated. I do this for 1991, and then I take the optimal portfolios derived for 1992 and repeat the process, continuing in this fashion for each of the years from 1991 through I then derive new optimal portfolios for the investor with 3, 5, and 10 year investment horizons by simply altering my mean-variance analysis to take into account the respective 3, 5, or 10 year Treasury Notes rate rather than the 1 year T-Bill rate. With these new portfolios, I analyze the 3, 5, and 10 year returns. These rates are taken as of April of the specified year, so as to be consistent with CT s study. This is the risk-free

35 29 rate that a potential investor would have used ex-ante. Thus, I am attempting to see if any single asset allocation technique outperforms the others, regardless of which year the investor entered the market, and regardless of how long of an investment horizon the investor possessed. The asset allocation methods are compared using Sharpe ratios, and are also compared against two benchmark portfolios: the minimum variance portfolio and the equally weighted portfolio. The minimum variance portfolio is used as a benchmark because, although the correlations are estimated well, the mean returns are not. The equally weighted portfolio is used as a naïve benchmark because in this allocation method, nothing is estimated well. If these portfolios outperform the portfolios based upon the model, then clearly analysts earnings forecasts provide no predictive power when used in conjunction with those valuation models. The Sharpe ratio, calculated by dividing the excess market return (the realized return above the risk-free rate) by the standard deviation of returns, is a measure of risk-reward. The higher the Sharpe ratio, the more return an investor gains per an equal amount of risk. Thus, asset allocation methods with higher Sharpe ratios are considered to be superior to those with lower Sharpe ratios.

36 30 VI. Results Table 4 details the optimal portfolios that were derived for each of the three methods, as well as the minimum variance benchmark. The inputs for the minimum variance portfolio are the same as for the historical portfolio, except that the expected returns for each index are set equal to one another (it does not matter what this value is). The various portfolios certainly are following different strategies. The historical based portfolios all tend to be heavily short on the DAX, extremely long on the Nikkei and FTSE, and in between for the S&P 500. Looking at Table 3, it becomes clear why. The DAX has had by far the poorest historical average returns, while at the same time the highest standard deviation. The reward-to-risk profile is extremely low compared to its international counterparts. Likewise, the Nikkei and FTSE have historically experienced much higher returns than the DAX or S&P, yet have standard deviations below the DAX and only slightly higher than the S&P 500. At the same time, CT s two models are heavily short on the DAX and the Nikkei and long on the FTSE and S&P 500. The implied expected rates of returns from their models differ from the historical mean returns. The expected rate of return from the abnormal earnings model for the DAX comes close to that of the historical average. However, Japan s implied rate of return is much lower and the United States is much higher. This interaction between the indices results in the heavy long-short positions in the two model based portfolios. The minimum variance portfolio, not surprisingly, is

37 31 comprised mostly of the Nikkei and the S&P 500, as they possess the two lower historical volatilities and are not nearly as correlated as the U.S. and U.K. The out-of-sample performances of the portfolios shown in Table 4 are listed in Table 6. These figures are arrived at by taking the product of the weight of the portfolio with the appropriate observed return as described in Table 5. The performances for the one year holding period are, not surprisingly, quite varied. For the and , and holding periods, the equally weighted naïve heuristic portfolio has higher achieved returns than either of the two model based portfolios. At the same time, those two portfolios, primarily due to their heavy leveraging, experience tremendously high returns in other years. Not coincidentally, these occur during years that the Nikkei, in which these portfolios hold short positions, achieved negative returns. For the one year holding period, the abnormal earnings model has an average one year return of 35.6% with a standard deviation of 30.9%. In contrast, the equally weighted portfolio has an average annual return of 11.52% with a standard deviation of 10.68%. For the longer holding period portfolios, the abnormal earnings model achieved higher returns than any of the other portfolios for every single holding period (3, 5, and 10 years) and regardless of which year the investor enters the market. The next highest returns are achieved by the dividend discount model. For example, if an investor had purchased a portfolio equivalent to the optimal portfolio determined by the abnormal earnings model in April of 1991, he or she would have achieved a 2,228.45% return, a compounded annual growth rate (CAGR) of nearly 37%. That is an astounding return. However, as was seen in the other holding periods, although the model based portfolios

38 32 allow for higher returns, they have much higher standard deviations. The heavy shorting elements in the portfolio composition allow for significant exposure to risk. The Sharpe ratio is a risk-adjusted measure of return. In order to calculate it, one needs to first subtract the risk free rate, compounded for the appropriate investment horizon, from the experienced return rates in order to get the excess return. The Sharpe ratio is the ratio between this excess return and the standard deviation of the returns for the given asset allocation method. Table 7 details the Sharpe ratio for each of the allocation methods for each holding period. Note that the historical portfolios all experienced negative Sharpe ratios for this period. This results from the fact that average excess return, which is the numerator of the ratio, was negative. That is to say, an investor would have experienced higher returns investing in solely U.S. government bonds than if they had used historical data to create portfolios.

39 33 VII. Discussion of Results The goal of this paper was to determine whether or not lowering your beliefs about the ex-ante equity premium would result in higher observed returns, as well as testing whether or not analysts earnings forecasts had any predictive power. Despite the fact that the data derived from CT s study were clearly superior inputs in this sample, the nature of mean-variance analysis tells us that it might not specifically have been the lower equity premiums that caused the superior returns. First of all, the reduction in the equity premia is not the driving force behind the superior returns from the two model based portfolios. Mean-variance analysis optimizes based on reward-to-risk profiles. Since the correlations coefficients between the various international markets are positive and high, ranging from 0.31 to 0.65, the main driver behind the portfolio weights is the difference in expected return between the asset classes. Mean-variance optimization is about relative rates of risk and return. For example, we could keep all of the model based data the same, except change Japan s average expected equity premium to 2% by raising our expected rates of return. This equity premium would still be significantly lower than the historical figures; however, the increase in expected return, along with the low historical level of risk in the Nikkei relative to the other 3 indices, would result in extremely long positions in the Nikkei as a part of portfolio. Observing the Nikkei s performance, one can see that the index lost a staggering 75% of its value from 1990 to 2002, and this portfolio would have performed as poorly as the ones in our sample did well.

40 34 Clearly, the superior results for the earnings based models come not from the fact that they have lower equity premia, but from the fact that embedded within those estimates was a bet that the Japanese market would not fare as well as the other markets. In fact, the extreme nature of the negative returns witnessed in the Nikkei in the 1990 s suggests that the index was extremely risky and probably should not have justified a lower equity premium. Although the shares of that index may have been overpriced, CT s model does not take this possibility into account, as their implied rate of return is equivalent to the discount rate which equates the present value of cash flows to the current market value. CT equate future cash flows to current market value. Thus, if the market valuation of a stock is high relative to its expected future earnings, there are two potential explanations. The first is that the discount rate used is too high. Lowering the discount rate increases the valuation using the DCF method. This is CT s resolution to the problem of high market valuations. However, the argument could be made that it is not the discount rate that is wrong, but that the stock itself is overvalued. Additionally, the mispricings could simply arise as a result of estimation error in analysts forecasts of future earnings. The analysts forecasts used by CT to derive their implied rates of return may in fact have predictive power. However, there are many variables to consider which suggest this is likely not true. The first is estimation error. With a small sample size, such as this case, in which relatively little data is available, there is much room for error. Unfortunately, estimating modern ex-ante equity premiums inherently result in a shortage of data. In addition, CT s models do not account for this error. They assume that current

41 35 market valuations are correct, and that when these valuations change, it is due to changes in the discount rate. Moreover, the variation in results from using the abnormal earnings model and the dividend growth model show that the effect that these earnings forecasts have on portfolio selection is dependent on the model that is used. As was stated earlier, mean-variance optimization revolves around relative reward-to-risk profiles. The goal of CT s paper was to justify a lower ex-ante equity premium, not test out-of-sample performance. Thus, these same earnings forecasts could have yielded much different relative equity premiums using alternative models, resulting in different portfolio selections and ultimately different achieved returns. There is also the issue of which perpetuity growth rates to use. Although they conducted a sensitivity analysis around their use of g 5 and g ae, the relative reward-to-risk ratios may have changed, even if the estimated equity premium did not change that much. Thus, more reasonable assumptions for these growth rates, rather than CT s optimistic assumptions, might have also yielded different optimal portfolio selections. Finally, as was discussed earlier, there are too many incentives for analysts to alter their forecasts to be able to draw a conclusion about its fundamental validity.

42 36 VIII. Conclusion Historical rates of return are frequently used in the derivation of mean-variance efficient optimal portfolios. However, recent research suggests that the historical equity premium may be too high, and ex-ante estimates place the future equity premium anywhere between 2 to 4%. The majority of the research surrounding the equity premium puzzle focuses on the determination of this actual value. However, the general investor cares more about the performance of these new methods as it pertains to achievable returns. In this paper, mean-variance efficient portfolios are constructed using three different sets of ex-ante expected returns: the historical return, an expected rate of return based on an abnormal earnings model, and an expected rate of return using a Gordon growth model. These portfolios are a combination of four risky assets, the major indices of Germany, Japan, the United Kingdom, and the United States. These efficient portfolios are then constructed at different periods with different holding periods and out-of-sample performances are calculated. As was expected, the portfolios created using the two models significantly outperformed those portfolios constructed using historical data. The use of the Gordon growth model allows for returns with much lower volatility than the use of the abnormal earnings model, while the latter offers a much higher upside potential. Surprisingly, the portfolios based on inputs from these models had much higher Sharpe ratios than those from two heuristic benchmarks: the minimum variance portfolio,

43 37 and the equally weighted portfolio. This suggests that there may be some predictive power behind analysts earnings forecasts, however, the small sample size does not allow for an investor to be able to draw that conclusion. Additionally, the simple embedding of lower expected equity premia into the mean-variance framework will not produce better performing portfolios. The relative reward-to-risk profiles of these assets are critical.

44 IX. Figures and Tables 38

45 39

46 40 Table 2: Model Based Expected Rates of Return This table details the implied expected rates of return for the four countries considered using the abnormal earnings and constant dividend growth models, as calculated by CT. The tables come directly from CT s paper. K is the expected rate of return assuming the abnormal earnings model, while k* is the expected rate of return assuming the dividend growth model. The implied rate of return is that discount rate which equates the valuation of the index with its market valuation. The five year earnings growth estimate, g 5, is the growth rate into perpetuity for the dividend growth model, and comes from analysts earnings forecasts. The growth rate into perpetuity for the abnormal earnings model is given by g ae. This value is set at the real risk free rate, which is approximated by the risk free rate minus 3%. If the risk free rate is below 3%, it is set at 0%. Thus, k-r f and k*-r f are the equity risk premiums implied by the abnormal earnings and dividend growth models respectively. The dividend growth model is given by equation (6) in the text and is detailed below. The abnormal earnings model is given by equation (10) in the text and is detailed below. Abnormal earnings is given by ae t = e t k(bv t-1 ), where e t is the earnings forecast for period t, bv t-1 is the book value of equity for the year ago period, and k represents the cost of equity charge. i! 1 5 (6) d1 *(1 + g) d1 d1 p0 =! k* = + g (10) aei ae5 ( 1 + gae) p [ ] * i * 0 = bv0 + " + i 5 ( 1 + k) ( k! g )(1 + k) = " i= 1 ( 1 + k ) k! g p 0 i= 1 ae

47 41

48 42