Excerpt

## Contents

1 Introduction

1.1 Studies about Sentiment

1.1.1 Sentiment by Surveys

1.1.2 Sentiment by Financial Variables

1.1.3 Sentiment by News

1.2 Research Questions

2 Empirical Methodology

2.1 Sentiment Data

2.2 Sentiment Index Computation

2.3 Incorporating Sentiment into the Portfolio

2.3.1 Blaek-Litterman Approach

2.3.2 Copula Opinion Pooling

2.4 Portfolio Optimization

2.5 Empirical Approach

2.6 Performance Measurement and Benchmarks

3 Results

3.1 Training Period Results

3.2 Test Period Results

3.3 Robustness Checks

4 Discussion

4.1 Sentiment Effect by Market

4.2 Sentiment Effect by Sign

4.3 Sentiment Effect by Business Cycle

5 Conclusion

A Appendix i

В References

## List of Tables

2.1 Summary of sentiment proxies

3.1 Performance measures for the training period

3.2 Performance measures for the test period

3.3 Performance measures for the test period when considering transaction costs

5,1 Positive sentiment and corresponding negative returns by market

## List of Figures

2.1 PC sentiment indices by market over time

2.2 Illustrating Markowitz’s Modern Portfolio Theory with an example , , , ,

2.3 Pairwise scatter plot for the asset return data

2.4 P-values of the likelihood ratio test on the shape parameter a

3.1 Cumulative return over time in the training period

3.2 Ratios of performance measures in the training period

3.3 Differences in performance measures in the training period

3.4 Cumulative return over time in the test period

3.5 Ratios of performance measures in the test period

3.6 Differences in performance measures in the test period

3.7 Ratios of Sharpe Ratios for different numbers of observations for the estimation of the initial market distribution

3.8 Ratios of Sharpe Ratios for different initial market distributions

3.9 3D plots of the Sharpe Ratios for varying parameter combinations of the view distribution

3.10 Heat maps of the Sharpe Ratios’ ranks for varying parameter combinations of the view distribution

3.11 Ratios of Sharpe Ratios for different specifications of the pick matrix , , ,

3.12 Ratios of Sharpe Ratios for different optimization techniques

3.12 Ratios of Sharpe Ratios for different optimization techniques (cont.) , , , ,

3.13 Empirical p-values of the sentiment bootstrap

4,1 Average weighted monthly return of the sentiment strategy by market ...

4.1 Average weighted return of the sentiment strategy by market (cont.) , , , ,

4.2 Average weighted monthly return of the sentiment strategy by market and sign of the respective sentiment index

4.2 Average weighted return of the sentiment strategy by market and sign of the respective sentiment index (cont.)

4.3 Monthly excess return of the sentiment over the initial market strategy by business cycle over time

4,4 Average monthly portfolio weight of the sentiment strategy by business cycle for each market

4,4 Average monthly portfolio weight of the sentiment strategy by business cycle for each market (cont.)

A.l PLS sentiment indices by market over time

A,2 Example illustrating the COP approach: market prior and posterior density by market

A.3 Differences in Sharpe Ratios for different numbers of observations for the estimation of the initial market distribution

A,4 Differences in Sharpe Ratios for different initial market distributions

A,5 Differences in Sharpe Ratios for different specifications of the pick matrix

A,6 Differences in Sharpe Ratios for different optimization techniques

A,6 Differences in Sharpe Ratios for different optimization techniques (cont.)

A,7 Ratios of Sharpe Ratios when allowing for short selling

A,8 Ratios of Sharpe Ratios when conditioning on extreme levels of sentiment

## List of Abbreviations

Abbildung in dieser Leseprobe nicht enthalten

## Abstract

In efficient financial markets, there is no room for sentimental investors. Any new information would be immediately absorbed and any mispricing immediately corrected by the forces of rational arbitrageurs doing the maths with the fundamentals. But why should finandal markets be different from any other market where humans interact and are subject to psychological biases (a long list of which is documented in Kahneman 2011)?

There is strong empirical evidence that investor sentiment, broadly defined as ‘a belief about future cash flows and investment risks that is not justified by the facts at hand’ (Baker and Wurgler 2007, p, 129), plays an important role in financial markets. It can lead to significant overpricing/underpricing, particularly of assets prone to subjective valuations. With limits/risks to arbitrage in the short term, prices rather correct over the medium to long term as sentimental beliefs mean-revert (De Long et al, 1990; Brown and Cliff 2005; Baker, Wurgler, and Y, Yuan 2012),

Building on the studies by Baker and Wurgler 2006 and Baker, Wurgler, and Y, Yuan 2012, measures of investor sentiment for international markets are constructed. Using the Copula Opinion Pooling approach developed by Attilio Meucci (Meueei 2006a; Meucci 2006b), this thesis shows how to incorporate these sentiment measures into portfolio optimization. Thereby, a sentiment-based trading strategy that exploits the medium-term reversal effect of sentiment is developed and empirically tested. The results are promising as they provide strong evidence that sentiment contains beneficial information that should not be neglected by quantitative portfolio managers.

## Chapter 1 Introduction

Are financial markets efficient?

Nobel prize winner Eugene Fama devoted his PhD work to the so called Random Walk Hypothesis (Fama 1965), He models the series of asset prices by a random walk, which means that the price increments are stochastic, more specifically, identically and independently distributed. This implies that historical price data has no ability to predict future prices. Empirically, using data from the Dow Jones Industrial Average stock market index, Fama 1965 finds support for his random walk hypothesis. Consequently, any change in market prices can only result from changes in fundamentals, for example, changes of a company’s earnings, liabilities or profitability. Consistent with Fama’s Efficient Market Hypothesis, this change would happen instantaneously as soon as the new information about fundamentals become available to the public. Therefore, given the assumption that all investors share the same set of information at all time, there is no room for sentimental investors in efficient markets.

But if there is no noise in information, then how can there be trading? Trade arises from disagreement what the price of an asset should be. As Black 1986, p, 531, argues, disagreement results from differing opinions about the future performance of an asset, which most likely reflects differing information or its differing use. Most famously, Shiller, Fischer, and Friedman 1984 object the efficient market hypothesis on the grounds of observations about social dynamics in the stock market that are consistent with psychological findings about overreaction. In fact, their data support the notion that social dynamics influence the stock market by enforcing price trends that contradict the assumption by Fama 1965 that the asset price time series can be modeled by a random walk. Nevertheless, Shiller, Fischer, and Friedman 1984 do not immediately reject the random walk hypothesis per se. Fama 1965, p, 94, himself acknowledges that some economic variables may have explanatory power in asset prices a posteriori. But the fact that these economic variables may be hardly predictable themselves makes them of little help to forecast future asset prices. In line with that, Shiller, Fischer, and Friedman 1984 conclude that the demand shocks due to social dynamics may also follow a random walk, since they are often difficult to anticipate.

What is sentiment?

In their seminal work, De Long et al, 1990 model sentiment as an overly optimistic or pessimistic view of so-called noise traders on financial markets. The term origins from Black 1986 and can be referred to as investors who trade on information that are not relevant to fundamental asset pricing. Following their views, these noise traders can collectively drive asset prices away from their fundamental values. Theoretically, this should open arbitrage opportunities for the so-called smart traders who do the maths, observe the mispricing and take the position opposite to the noise traders. When the mispricing becomes apparent, asset prices return to their fundamental value, and the smart traders gain the initial price difference in return.

Surprisingly though, De Long et al, 1990 show that, under certain conditions, noise traders induced over- or underpricing will not be arbitraged by smart traders in the short term. The reason for this is that noise traders may drive up prices to even greater extremes (De Long et al, 1990, p, 727), before they realize their “irrational exuberance” (Greenspan 1996).

Now, if there are more smart than noise traders in the market, they could still bring back prices to their intrinsic value if they collectively traded against the noise traders. However, it can be that a) the smart traders are in the minority, or b) the majority is risk-averse because they think prices will not revert in the short term, and therefore, they will earn negative returns to the point where holding the opposite position becomes too costly. Regarding a), Shiller, Fischer, and Friedman 1984, p, 497-498, note that short-term arbitrage profits are often limited due to a lack of smart traders in the economy. Regarding b), Brown and Cliff 2005, p, 409, note that avoiding noise trader risk is reasonable because many professional fund managers are evaluated annually. Thus, it might already be too late if the opposite-to-noise-trader-position pays off after the manager’s evaluation.

How can smart traders exploit sentiment?

It seems that if smart traders are willing to exploit the mean-reversion effect of sentiment, their best strategy essentially becomes a matter of the right market timing. De Long et al, 1990, p, 727, specifically motivate this study:

In a world with mean-reverting noise traders’ misperceptions, the optimal investment strategy is very different from the buy and hold strategy of the Standard investment model. The optimal strategy for sophisticated investors is a market timing strategy that calls for increased exposure to stocks after they have fallen and decreased exposure to stocks after they have risen in price. The strategy of betting against noise traders is a contrarian investment strategy: it requires investment in the market at times when noise traders are bearish, in anticipation that their sentiment will recover.

Exploiting investor sentiment for portfolio optimization Of course, the best market timing of the contrarian sentiment-based investment strategy depends on having a precise measure of sentiment in the first place. Sections 1,1,1, 1,1,2 and 1,1,3 present a host of studies about different kinds of sentiment measures, as well as empirical findings about their effects on the stock market. Based on these findings, the precise research hypotheses of this thesis are formulated in Section 1,2,

As the starting point of the empirical investigation, this thesis uses the well-known sentiment proxies found in Baker and Wurgler 2006 and Baker, Wurgler, and Y, Yuan 2012 to construct monthly sentiment indices for three large international markets, namely Europe (EU), Japan (JP) and the United Kingdom (UK), For the United States (US), it uses the sentiment index by Baker and Wurgler 2006, Furthermore, a global sentiment index is constructed from the four regional sentiment indices. The studies by Baker and Wurgler 2006 and Baker, Wurgler, and Y, Yuan 2012 are also reviewed in Section 1,1,2, and the proxies and sentiment indices are presented in detail in Section 2,1 and 2,2,

The goal of this study is then to develop a trading strategy that exploits the information contained in investor sentiment. This goal is achieved by demonstrating the innovative use of a method called Copula Opinion Pooling (COP) by Attilio Meticci (Meueei 2006a; Meticci 2006b) to incorporate investor sentiment into portfolio optimization. The COP methodology solves shortcomings of the more renowned Blaek-Litterman (BL) methodol- ogv (Black and Litterman 1992), Both methodologies are theoretically explained in Section 2,3, Whereas the BL approach provides analytical formulae to compute the optimal portfolio weights, the COP approach is a numerical method. It simulates market scenarios which can be used for arbitrarily portfolio optimization techniques. The most renowned technique is shown in Section 2,4 and optimizes the mean-variance trade-off following the Modern Portfolio Theory by Henry Markowitz,

In line with explanatory findings, Section 2,5 describes the specifications of the sentimentbased trading strategy and, from February 1987 to September 2015, estimates the parameters from a dataset of stock market data of the four major stock market indices corresponding to the aforementioned regional sentiment indices, i.e, the EUEOSTOXX50 (EU), the NIKKEI225 (JP), the FTSE1000 (UK) and the S&P500 (US), as well as the US Dollar Gold Price per troy ounce (GLD) which is inversely related to the global sentiment index,^{1} After developing the sentiment strategy, Section 2,6 establishes the performance measures and the benchmarks against which the sentiment strategy is compared.

The results in Chapter 3 show if sentiment contains information that is meaningful and exploitable by the COP methodology. Because the parametrization of the sentiment strategy is not unambiguous, the dataset is split equally into a training (Section 3,1) and test period (Section 3,2), which both cover more than one business cycle. By splitting the data, it can be assessed if the initial parameters lead to an equivalent performance in both sample periods. To further cheek if the specification is not sensitive to small changes in the initial parameters, Section 3,3 involves a battery of robustness cheeks.

Finally, Chapter 4 discusses the results and sheds light on some plausible concerns, i.e, that the sentiment strategy’s performance may be attributed to a certain market only (Section 4,1), that the sentiment effect is one-sided (Section 4,2) or that it is dependent on the state of the economy (Section 4,3), Chapter 5 concludes with a short summary of the paper in response to the research questions stated in Section 1,2, and it presents topics for further research as well as implications of this study for portfolio management.

The following section commences with an overview of the sentiment literature,

### 1.1 Studies about Sentiment

A large stream of literature on the effects of investor sentiment emerges after the seminal work by De Long et al, 1990, They use an overlapping generations model to analyze the effects of noise traders on the stock market. Their model rejects the assumption that any noise trader induced mispricing will immediately be arbitraged by sophisticated investors. The main reason for this is that noise traders create risk in the sense that it is difficult to predict when noise traders will eventually change their mispereeived beliefs about asset prices. This in turn makes it hard to anticipate when prices will revert to their fundamental value. Even more striking, De Long et al, 1990 show that, under certain conditions, noise traders can earn even higher returns than their rational counterparts. Although the probability for this scenario is rather low, it clearly illustrates the potential effect that noise traders can have on financial markets.

In a more recent theoretical study, Fu, Jacoby, and Wang 2015 also emphasize the importance to consider investor sentiment in portfolio selection. They separate the return into a fundamental and sentiment-related error component. They show that irrational prie- ing and increased market volatility alter the mean-variance trade-off in the Capital Asset Pricing Model (CAPM), Consequently, any professional investor neglecting the effect of noise traders on the first two moments, mean return and variance, is at risk of selecting a sub-optimal portfolio (Fu, Jacoby, and Wang 2015, p, 272), This result is particularly important to this study because this study compares three strategies that all maximize the mean-variance trade-off while only one, the sentiment strategy, takes into account investor sentiment.

There is an abundance of literature about measures of sentiment and their empirical effects on the risk and return characteristics of various asset classes,^{2} To structure the content, each study was classified by the type of measurement: surveys (Section 1.1.1), financial variables (Section 1.1.2) and news (Section 1.1.3).

De Long et al. 1990 hypothesize that while sentiment-induced mispricing may not be exploitable in the short term, the mean-reversion of sentiment may still provide arbitrage opportunities on longer time horizons. That is why the review of the subsequent empir- ieal findings will also devote considerable attention to the time horizon of sentiment effects.

#### 1.1.1 Sentiment by Surveys

A logical starting point to measure investor sentiment is simply to ask investors about their market expectations. Two commonly used investor sentiment surveys are the US survey by the American Association of Individual Investors (AAII) and the survey by Investors Intelligence (II). The AAII survey asks its members each week since 1987 if they believe the stock market will go up (bullish investors), not change (neutral investors) or go down (bearish investors) over the next months. The II survey does not ask investors directly. Instead, it extracts investor sentiment by classifying market newsletters as bullish, bearish or neutral. Since II surveys market newsletters which are more often written by market professionals, the II survey more likely tracks institutional investor sentiment, whereas the AAII survey tracks individual investor sentiment (Brown and Cliff 2004, p. 7).

Abbildung in dieser Leseprobe nicht enthalten

Typically, the sentiment in investor surveys is seen as a contrarian indicator of the future stock market performance as most investors are just trend followers willing to pay unusually high (low) prices when they are bullish (bearish) (Burke 2002). However, an analysis by Eotblut 2013 suggests that the effect is conditional on the level of optimism/pessimism, i.e. the more extreme the level of sentiment, the more reliable is the contrarian indicator. There are various ways to compute a sentiment measure from the sentiment survey data, e.g. the share of bullish or bearish sentiment. A measure that is commonly used in literature is the bull-bear-spread (BSS). It is computed as where BULLISHt is the number of investors being bullish in week t, respectively NEUTRALt the number being neutral and BEARISHt the number being bearish. Brown and Cliff 2004 compute the BSS for both the AAII and II survey to compare the effects of individual and institutional sentiment on the US stock market. First, they find that sentiment by surveys, which is a direct measure of sentiment, shares a common factor with other indirect measures of sentiment, such as the elosed-end fund discount (CEFD) or the number of initial public offerings (NIPO) (see Section 1.1.2 for the review of indirect measures). Extracting this common factor, they create a composite measure of sentiment and explore stock market effects in the short term. They find no strong evidence that sentiment predicts stock market returns in the short term of up to 2 months (Brown and Cliff 2004, see Table 4), The strongest link can be seen between institutional sentiment and the returns of large stocks (Brown and Cliff 2004, see Table 6), The latter finding contrasts the conventional wisdom that individual investors are more subject to sentiment. Since individual investors predominantly hold small stocks, one would rather suspect a sentiment effect of individual investor sentiment on the returns of small stocks (Lee, Shleifer, and Thaler 1991),

Using only the II survey, Brown and Cliff 2005 investigate the hypotheses that excessive optimism leads to market overvaluation (HI), which is followed by subsequent low market returns when prices revert to their intrinsic value in the long term (H2), Regarding HI, they use the pricing errors from Bakshi and z, Chen 2005 on most stocks in the Dow Jones Industrial Average, They can confirm that sentiment leads to mispricing by affecting the market price, but not the intrinsic price,^{3} Regarding H2, they use long-horizon regressions and, between 1963 and 2000, find a strong negative relationship between sentiment and market returns over the next 1 to 3 years (Brown and Cliff 2005, see Table 3), The effect is also negative for 6 months, but less significant. This is in line with the theory that it may take some time before investors change their believes, or before rational arbitrageurs force back prices to their intrinsic value. As in Brown and Cliff 2004, the effect is stronger for large stocks. They attribute this result to the fact that II surveys market newsletters that consider the market as a whole and are therefore more focused on larger stocks (Brown and Cliff 2004, p, 422), Furthermore, the effect is stronger on growth stocks, which is reasonable because growth stocks are more difficult to value and therefore more sensitive to sentiment as a wider range of valuations are justifiable (Baker and Wurgler 2006, p, 1646),

Generally, any effect on stock returns found for some variable should survive the inclusion of the widely accepted three asset pricing factors by Fama and French 1993 (FF3F), The Fama-French 3 Factor Model (FF3M) is an improvement over the classical Capital Asset Pricing Model (CAPM), It explains common variation in the eross-seetion of stock returns. The FF3F are:

**- RMRF** (market return minus risk-free rate): This is the equivalent to the classical CAPM market risk factor,

**- SMB** (small minus big): Small stocks have higher expected returns on average than large stocks,

**- HML** (high minus low): Stocks with higher book-to-market value have higher ex- peeted returns on average,

Thus, in the FF3M, the stock return r can be expressed as

Abbildung in dieser Leseprobe nicht enthalten

Fama and French 2015 extend the model to a five factor model that additionally considers profitability and investment. There is extensive literature on additional factors.^{4}

The findings by Brown and Cliff 2005 are robust to the inclusion of the FF3F and a series of other asset pricing variables. Sehmeling 2007 builds on the work by Brown and Cliff 2005 and also employs long-horizon regressions to test for a sentiment effect on international stock markets up to 75 weeks ahead. His dataset covers the DAX30, the EUROSTOXX50, the NASDAQ100, the S&P500 and the NIKKEI225, Sentiment is measured by the BSS of the Sentix survey. The survey started in 2001 and asks both institutional and individual investors about their short- and medium-term market expectations. Considering only the medium-term answers, Sehmeling 2007, see Figure 2, shows that institutional sentiment positively and individual sentiment negatively predict stock market returns on most horizons. This result is in line with the argument that institutional sentiment proxies for smart money, whereas individual sentiment proxies for noise trader risk. Another interesting finding by Sehmeling 2007 is that institutional investors seem to anticipate and align their strategy to the over-optimism/pessimism by individual investors.

A point of criticism regarding the results by Sehmeling 2007 is that his data only ranges from 2001 and 2006. It therefore covers less than one business cycle. He even acknowledges that there is a structural break in the data (Sehmeling 2007, see Figure 1). Also, the negative effect of individual sentiment is not significant in the second half of the period and the influence of individual sentiment on institutional sentiment fades. Sehmeling 2007, p. 142, interprets this break as a sign of individual investors starting to follow the smart money.

Also Heiden, Klein, and Zwergel 2013 and Schneller et al. 2017 use the sentix survey as a measure of sentiment, but use it to look at effects other than on market returns. Heiden, Klein, and Zwergel 2013 build on the work by Menkhoff and Rebitzky 2008 who find that sentiment predicts exchange rate movements in the long term of over two years. Heiden, Klein, and Zwergel 2013, see Table 2, show that institutional sentiment can be used to forecast the Euro/US Dollar exchange rate between around 8 to 24 weeks, but not the Japanese Yen/us Dollar exchange rate. Since most respondents to the Sentix survey are European, the respondents are probably in a better position to assess the Euro/us Dollar exchange rate rather than exchange rates with two foreign currencies (Heiden, Klein, and Zwergel 2013, p, 11),

Schneller et al, 2017 provide further evidence for a home bias, which is a well-known phenomenon in international equity markets (French and Poterba 1991), Measuring the dispersion in sentiment from the Sentix survey, Schneller et al, 2017 find that it is only related to expected stock market volatility in Europe, but not in Japan or the US,

Besides investor surveys, there is also a large stream of literature that employs different surveys which are assumed to proxy for sentiment regarding asset prices. For example, Lemmon and Portniaguina 2006 use the consumer confidence indices by the Conference Board and by the University of Michigan, Consumer confidence should rather proxy for individual than for institutional investor sentiment. Indeed, Lemmon and Portniaguina 2006, see Table 3, show that small stocks earn lower returns following quarters of high eonsumer confidence (individual sentiment). However, the effect is only present in the period between 1978 and 2002, but not in the pre-1977 period. They attribute this to the fact that the importance of individual investors in the stock market has increased over time, which means their impact on asset prices was not strong enough in the earlier period (Lemmon and Portniaguina 2006, p, 1502),

Sehmeling 2009 extends the analysis by Lemmon and Portniaguina 2006 and investigates the effect of consumer confidence on the stock market in 18 industrialized countries. On average, there is a negative sentiment effect for horizons of up to 24 months (Sehmeling 2007, see Table 4) and there is eross-seetional variance in the effect that can be attributed to varying levels of market integrity and cultural differences (Sehmeling 2009, see Table 8), The latter means that there is a stronger sentiment effect in countries that are less indi- vidualistie. This is in accordance with the finding by Chui, Titman, and Wei 2010 that momentum profits are higher in countries where investors follow more the consensus than their own opinion. This in turn is consistent with the argument made in Shiller, Fischer, and Friedman 1984 that social dynamics play an important role in financial markets,

#### 1.1.2 Sentiment by Financial Variables

A frequent point of criticism about surveys is that the respondents do not act according to their opinion that they share in the survey. A famous saying in the financial world is to “put your money where your mouth is”. That justifies the search for financial variables that proxy for the “acting” part of investor sentiment,

Neal and Wheatley 1998 use three financial market variables as proxies for individual investor sentiment: the elosed-end fund discount, net mutual fund redemptions and the odd-lot ratio of sales to purchases. They find that elosed-end fund discounts strongly and net mutual fund redemptions weakly predict the return of small US stocks over multiple horizons of up to four years between 1933 and 1993, Note that a discount (premium) on closed-end funds arises when they are trading below (above) their net asset value. Thus, elosed-end fund discounts and redemptions from mutual funds reflect negative sentiment, whereas premiums and inflows reflect positive sentiment. The positive sentiment effect in Neal and Wheatley 1998 is therefore consistent with the theory that low (high) sentiment is a buying (selling) indicator. This indicator should be more informative for small stocks which, according to Lee, Shleifer, and Thaler 1991, are predominantly held by individual investors.

The empirical study in this thesis centers on the sentiment proxies used in Baker and Wur- gier 2006 and Baker, Wurgler, and Y, Yuan 2012, Baker and Wurgler 2006 use principal component analysis (PCA) to construct a sentiment index for the US out of six financial variables which they believe should proxy for investor sentiment,^{5} The underlying assumption is that all six proxies are driven by a common sentiment factor which is captured by the first principal component (PC), Thus, principal component analysis helps to eliminate the variation in the proxies that is not driven by sentiment,^{6} The sentiment proxies Baker and Wurgler 2006 consider are:^{7}

**- Closed-end fund discount (CEFD):** The higher the discount, the lower is sentiment (Neal and Wheatley 1998),

**- Market turnover (TV):** Turnover increases with investor optimism,

**- Number of IPOs** (NIPO): The best timing for IPOs is when investor sentiment is high (Ritter 1991),

**- Return on IPOs:** On average, higher first-day returns on IPOs are expected when investors are bullish (Ritter 1991),

**- Equity share in new total debt and equity issues:** More equity is issued when sentiment is high and prices are overvalued (Baker and Wurgler 2000),

**- Dividend premium:** Investor sentiment leads to shifts in demand for dividend paying and non-paving stocks. This causes firms to pay (not pay) dividends when investor demand implies a dividend discount (premium) on dividend paying stocks (Baker and Wurgler 2004),

Using their US sentiment index (BWUSSI), Baker and Wurgler 2006 find that, between 1963 and 2001, high subsequent market returns are followed by periods of low sentiment, but vice versa the effect is not significant. The negative sentiment effect is stronger, the more extreme the sentiment (Baker and Wurgler 2006, p, 1665), When sorting stocks into deciles by characteristics such as growth or book-to-market value, Baker and Wurgler 2006, see Table III and Figure 2, reveal a U-shaped pattern which means that the effect of sentiment is more pronounced for firms in both extremes, e.g. firms experiencing high growth or firms that are in financial distress. Although the sentiment effects are attenuated when controlling for the FF3F, they stay significant (Baker and Wurgler 2006, p, 1671), Furthermore, the effect of each sentiment proxy individually has the expected sign, though the effect sizes vary: the elosed-end fund discount and the number of IPOs are better predictors (Baker and Wurgler 2006, p, 1671),

Baker and Wurgler 2007 extend the study by Baker and Wurgler 2006 and compute a sentiment change index based on the first PC of the changes in the sentiment proxies to estimate sentiment betas. They find that stocks that are more difficult to arbitrage and therefore subject to more speculative investments also have higher sentiment betas (Baker and Wurgler 2007, see Panel в of Figure 4), Regarding small stocks, Baker and Wurgler 2007, p, 148, note that the sentiment effect on the aggregate market level is stronger when considering equally weighted rather than market cap weighted returns.

Baker and Wurgler 2006 actually use the version of their sentiment index that is also orthog- onalized to several macroeconomic variables such as the growth in the industrial production index or the recession indicator of the US National Bureau of Economic Research (NBER), Neither the index nor the results change significantly when using the unorthogonalized versión (Baker and Wurgler 2006, p, 1665), This result alleviates concerns that the BWUSSI is rather a proxy for the condition of the overall economy than for the irrational eompo- nent of sentiment. However, the findings by Sibley et al, 2016 question whether the macro variables used for orthogonalization may adequately capture the business cycle. According to their results, 63% of the total variation in the BWUSSI can be explained by 13 risk and business cycle variables. When separating the index into the business cycle and a residual component, they show that only the business cycle component has significant predictive power (Sibley et al, 2016, see Table 4), That the residual component is unrelated to market returns is problematic given the definition in Baker and Wurgler 2007, p, 129, of investor sentiment as “a belief about future cash flows and investment risks that is not justified by the facts at hand,” Here, risk and business cycle variables should count as facts that are considered by rational investors.

But generally, is the assumption that sentiment is completely irrational not too restrictive? Qiu and Welch 2004, p, 30, make a good point about this:

[,,,] we would expect that less unemployment and more wealth and a rising stock market would make naive retail investors more buoyant about the future, (In what ‘[behavioral]’ theory would this not be the ease?) Optimism and sentiment that is correlated must not fall like manna from heaven. If some proposed proxy does not correlate with any macroeconomic conditions, e.g,, employment, recessions, and depressions, it would not exactly speak for this measure as a proxy for sentiment.

Thus, in contrast to the noise trader theory developed by De Long et al, 1990, sentimental investors may not entirely trade on noise. However, it is still possible that they overreact because, for example, they are too confident about the value or implications of their information, a psychological bias which is documented in Daniel, Hirshleifer, and Subrahmanvam 1998, On the other hand, Sibley et al, 2016, p, 178, acknowledge that their findings show correlation between sentiment and risk and business cycle variables, but not causality. It is equally possible that sentiment affects the fundamentals which then transmit the effect onto the stock market. Indeed, Stambaugh and Y, Yuan 2017 show that the BWUSSI predicts fundamental pricing factors which are related to well-known stock market portfolio anomalies.

Subsequent empirical studies show that the BWUSSI is a contrarian indicator for the performance of stocks. For example, Stambaugh, Yu, and Y, Yuan 2012 explore how sentiment affects the performance differential in stock market anomalies. In order to do that, they sort stocks into deciles according to 11 well-documented anomalies adjusted for the FF3F,^{8} They then generate long-short strategies which go long the stocks in the top decile and short the stocks in the bottom decile.

First and foremost, Stambaugh, Yu, and Y, Yuan 2012, see Table 2, show that, between 1965 and 2007, the performance differential in each anomaly is stronger following high levels of investor sentiment as measured by the BWUSSI, This result is somehow different from the finding by Baker and Wurgler 2006 that only low sentiment is significantly related to market returns, Stambaugh, Yu, and Y, Yuan 2012 reason with Miller 1977 that overpricing is more prevalent than underpricing due to short-sale impediments. That is, a minority of over-optimistic investors can bid up prices to excessive values because their well-informed counterparts often rather prefer to take no position than to take a short position.

Second, consistent with the previous argument, the results by Stambaugh, Yu, and Y, Yuan 2012 show that high sentiment primarily affects the short decile, where excessive optimism leads to overpricing which is followed by lower returns. Thus, the long-short strategy earns higher profits mostly because it is short in bottom decile, whereas the long decile has fewer potential to increase the spread because underpricing is less prevalent, Stambaugh, Yu, and Y, Yuan 2014 add robustness to the results found in Stambaugh, Yu, and Y, Yuan 2012, Their work addresses the concern that the BWUSSI is a spurious predictor. They simulate over 200 million spurious sentiment regressors and report that the number of simulations to find at least one regdstock market anomalies as the BWUSSI is very high (Stambaugh, Yu, and Y, Yuan 2014, see Table 1), Consequently, it is very unlikely that the relationship between sentiment and the stock market anomalies is spurious.

Similarly to Stambaugh, Yu, and Y, Yuan 2012, Chung, Hung, and Yeh 2012 form longshort portfolios on 11 stock market anomalies and separate the state of the economy by the NBEE recession indicator and, alternatively, using a Markov-switehing model. They find that the sentiment effect is only present during expansionary periods, where high levels of the BWUSSI are followed by lower market returns, particularly for stocks that are more subject to sentiment.

Finally, Yu and Y, Yuan 2011 use the BWUSSI to binary classify sentiment periods. They find that the mean-variance trade-off is altered during high sentiment periods. In essence, their results suggest that when sentiment is high, investors tolerate lower compensation for the same amount of risk as compared to when sentiment is low. Hence, Yu and Y, Yuan 2011 provide empirical evidence for the theoretical result in Fu, Jacoby, and Wang 2015 that investors should incorporate investor sentiment into portfolio optimization.

Given the strong empirical evidence that the BWUSSI is a good predictor of the US stock market performance, it is a reasonable next step to extend the study by Baker and Wurgler 2006 to international markets.

Baker, Wurgler, and Y, Yuan 2012 use PCA to construct country-level sentiment indices for Canada, France, Germany, Japan, the UK and the US, Three of their four sentiment proxies - the number of IPOs (NIPO), the average first-day return on IPOs and market turnover (TV) - are also part of the set in Baker and Wurgler 2006,^{9} The fourth sentiment proxy - the volatility premium (VP) - is highly related to the dividend premium used in Baker and Wurgler 2006, which Baker, Wurgler, and Y, Yuan 2012 could not obtain for all markets.

Baker, Wurgler, and Y, Yuan 2012 also construct a global sentiment index as the first PC of the country-level indices. They orthogonalize the country-level indices to this global index to obtain local sentiment indices.

Regarding the results, Baker, Wurgler, and Y, Yuan 2012, see Figure 1, first of all observe that the country-level sentiment indices exhibit mean-reversion, which is in line with the theory about noise traders changing their exaggerated beliefs over the medium to long term (here about 1 to 2 years (Baker, Wurgler, and Y, Yuan 2012, p, 281)), As a consequence, country-level sentiment, which is mostly driven by global sentiment, is a contrarian indicator of overall market performance (Baker, Wurgler, and Y, Yuan 2012, see Table 5), Within a market, local sentiment can significantly explain eross-seetional differences (Baker, Wurgler, and Y, Yuan 2012, see Table 7), They attribute this pattern to the fact that international investors are mostly investing on the aggregate level to seek diversification (Baker, Wurgler, and Y, Yuan 2012, p, 285), Therefore, the aggregate market is more subject to global sentiment. On the other hand, local investors have a home bias towards their local market (French and Poterba 1991), This explains why eross-seetional variation in returns is more subject to local sentiment (Baker, Wurgler, and Y, Yuan 2012, p. 285).

Interestingly, Baker, Wurgler, and Y, Yuan 2012 also show that sentiment is contagious. By regressing eross-seetional returns on local sentiment, the US sentiment index and its interaction with capital flows, they find that US sentiment spreads internationally through capital flows (Baker, Wurgler, and Y, Yuan 2012, see Table 8),

A major drawback of the study by Baker, Wurgler, and Y, Yuan 2012 regarding its immediate applicability to the empirical study in this thesis is the low annual frequency of the international sentiment indices as opposed to the monthly BWUSSI, It is striking that even with this small number of just 25 yearly observations, the results are significant and robust to the inclusion of the FF3F, This motivates the empirical study in this thesis, which combines proxies and the BWUSSI from Baker and Wurgler 2006 with proxies from Baker, Wurgler, and Y, Yuan 2012 to provide five monthly international sentiment indices over a period of more than 20 years,

#### 1.1.3 Sentiment by News

Sentiment by news has mainly evolved over the last decade. Broadly defined, sentiment by news identifies sentiment that is indirectly expressed through various channels such as news articles, social media or search queries.

For example, Tetloek 2007 extracts sentiment from the daily content in a popular column of the Wall Street Journal between 1984 and 1999, He quantifies sentiment as the first PC on a set of word frequencies by 77 predefined categories of the Harvard psychological journal. The first PC is found to be related to media pessimism, Tetloek 2007 can show that higher media pessimism is followed by negative stock returns on the first day. However, this initial downward price movement is fully reversed over the next four days (Tetloek 2007, see Table II), Therefore, he concludes that the media column does not provide new information that is not yet reflected in the fundamental asset price as otherwise the negative effect would persist. Instead, the reversion indicates that the media column triggers a sentiment effect (Tetloek 2007, p, 1150),

H, Chen et al, 2014 extract sentiment from a popular social media platform for investors in the US called “Seeking Alpha”, They measure sentiment as the fraction of negative words in articles and comments posted on the website. Their results are encouraging to investors seeking advise from their peers as both sentiment expressed in articles and comments prediet stock market returns in the direction of sentiment, H, Chen et al, 2014, p, 1370-1371, also discuss why investors are seemingly providing informational advantage to their peers: aside from feedback and recognition, popular authors can also receive financial rewards, as well as potentially benefit from a bandwagon effect if readers follow their recommendations, Sul, Dennis, and L, Yuan 2017 consider another social media platform and extract sentiment from tweets on Twitter about firms in the S&P500, Interestingly, they find that users with fewer followers and whose tweets are not re-tweeted have a stronger effect on future stock returns than re-tweeted tweets of users with more followers. Their interpretation of this seemingly counterintuitive finding is related to market efficiency as the authors believe that tweets reaching a greater audience spread information more quickly and broadly so that the sentiment effect of these tweets is immediate (Sul, Dennis, and L, Yuan 2017, p, 475), On the other hand, a slower diffusion of information by less popular tweets corresponds to a sentiment effect that is observable on longer horizons.

Finally, Da, Engelberg, and Gao 2015 measure sentiment from the search volume on Google for economic terms that express negative sentiment. They create a sentiment index comprising of a set of 30 terms that are dynamically selected by looking backwards and identifying the terms that were the best predictors of future market returns. The authors argue that their “Financial and Economic Attitudes Revealed by Search (FEARS) index” is more beneficial than sentiment surveys because a) the FEARS index is available at a higher (daily) frequency, and b) search volume potentially is more revealing about the intentions and information of a person than a response in a survey (Da, Engelberg, and Gao 2015, p, 4-5), They find that an increase in FEARS is associated with contemporaneously low stock market returns, but high treasury returns (Da, Engelberg, and Gao 2015, see Table 2 and 3), However, this effect reverses with stocks earning higher returns than treasuries over the next two days. In line with previous findings, the sentiment effect is stronger for stocks more prone to speculation, e.g, small and high-volatile stocks as well as stocks with high market beta and greater downside risk (Da, Engelberg, and Gao 2015, see Table 4),

The emergence of more efficient tools for textual or search query analysis gives rise to a new set of sentiment measures that presumably contain more revealing information about investor sentiment than surveys. Also, they are probably better proxies for the irrational component of sentiment than financial proxies blurred by fundamentals. However, there are two major drawbacks limiting their application towards a robustly tested sentiment-based trading strategy. First, most of the measures have not underwent the amount of review and testing as for example the BWUSSI did. Second, up to now, sentiment measures based on news often lack a history long enough to allow for serious backtesting of trading strategies that are required to deliver outperformanee over the long term that includes multiple business cycles. For example, the FEARS index starts in 2004, whereas the BWUSSI already starts in 1965, That is why for the purpose of developing a sentiment-based trading strategy, sentiment measurement is based on the proxies in Baker and Wurgler 2006 and Baker, Wurgler, and Y, Yuan 2012 in this study ,

The following section states the research questions of this thesis,

### 1.2 Research Questions

Measuring sentiment as in Baker and Wurgler 2006 and Baker and Wurgler 2007, this thesis aims to develop a sentiment-based trading strategy to empirically test if sentiment can be exploited for portfolio optimization. Although a series of explanatory findings suggest that sentiment plays an important role in financial markets and sentiment-based trading strategies can earn superior returns^{10}, sentiment has so far rarely been used in portfolio optimization. This is even more surprising as the theoretical work by Fu, Jacoby, and Wang 2015 and the empirical evidence by Yu and Y, Yuan 2011 imply that sentiment alters the mean-variance trade-off. Investors are therefore advised to consider sentiment for their portfolio decisions,

Attilio Meueei (Meueei 2006a; Meueei 2006b) presents a flexible way to blend investors views with the market prior. But also his Copula Opinion Pooling (COP) approach has so far rarely been used in practice (see Section 2,3,2), The first research question of this thesis therefore is:

Q!: How can investor sentiment be integrated into portfolio optimization using the COP methodology?

Section 2,5 demonstrates how the sentiment indices computed in Section 2,2 can be ineor- porated into the portfolio optimization procedure. To empirically assess if the sentiment- adjusted portfolio earns higher stock market returns, the second research question asks:

Q2: Does investor sentiment provide beneficial information for portfolio optimization?

If that is true, then an optimized portfolio that takes sentiment into account should outperform a portfolio that neglects it. Quantitatively, Chapter 3 compares the risk-adjusted profits of the sentiment strategy against a strategy that simulates from the same market prior, but without considering sentiment. To ensure that the overall good performance cannot solely be attributed to the COP methodology, several other frequently used trading strategies complete the set of benchmarks, A series of robustness cheeks examines if the performance of the sentiment strategy is sensitive to the parametric specification. Furthermore, in a way similar to Stambaugh, Yu, and Y, Yuan 2014, a bootstrapped procedure is used to make sure that sentiment provides incremental information and its effect is not some sort of statistical artifact.

De Long et al, 1990, p, 727, suggest that the optimal sentiment strategy is a contrarian market timing strategy. Based on the empirical findings outlined in Section 1,1, the third research question addresses the optimal timing of investing contrarily to current investor sentiment,

Q3: Does a medium to long term investment horizon exploit the mean-reversion of sentiment-induced mispricing optimally?

The mean-reversion of sentiment is well-documented in the literature (see for example Baker, Wurgler, and Y, Yuan 2012, Tetloek 2007 or Da, Engelberg, and Gao 2015), De Long et al, 1990 predict that betting against investor sentiment in the short term may be very costly because noise traders may not change their beliefs immediately and continue to bid up prices to even greater extremes. Consistent with the theory of limits to arbitrage in the short term, Brown and Cliff 2005 find a more significant negative effect of sentiment on market returns for medium to long-term investment horizons. Thus, it is expected that a sentiment strategy overweighting (underweighting) a market over the medium to long term when market sentiment has previously been low (high) should exploit sentiment-induced mispricing optimally.

## Chapter 2

### Empirical Methodology

Chapter 1 discussed various findings about the effects of investor sentiment on financial markets. However, these findings are mostly explanatory in the sense that the relationship is empirically tested, but less often exploited with a sentiment-based trading strategy,^{11} A reason for this might be the difficulty to translate the measured sentiment into portfolio decisions.

This chapter discusses two mathematical frameworks to incorporate sentiment into portfolio optimization. The sentiment is based on five indices which are derived from respectively four financial measures that proxy for investor sentiment. These measures are introduced in Section 2,1, Section 2,2 shows how to build the sentiment indices from them. Section 2,3 explains the two mathematical frameworks to incorporate investors’ views/sentiment into portfolio optimization. The Copula Opinion Pooling approach used in this study does not prescribe a specific optimization approach. Since the focus of this study is to empirically examine whether investor sentiment can be exploited for portfolio optimization, a classi- eal mean-variance optimization is performed, which is explained in Section 2,4, The main contribution of this thesis is centered around Section 2,5, which describes the empirical setup and shows how the sentiment effect on the portfolio is modeled. Finally, Section 2,6 presents the measures and benchmarks, which are used to set the performance of the developed sentiment strategy into context. The results are shown in the following Chapter 3,

### 2.1 Sentiment Data

This section introduces the sentiment measures that are used to construct the sentiment indices in Section 2,2,

Due to data limitations, the choice of measures often determines the portfolio and vice versa. As justified in Section 1,1,2, the choice of sentiment measures is based on the work by Baker and Wurgler 2006 and Baker, Wurgler, and Y, Yuan 2012, The former construct a monthly sentiment index for the us market (BWUSSI) based on six variables which should proxy for investor sentiment. However, most of these proxies are difficult - if not impossible - to obtain for other stock markets over a long period of time. That is why Baker, Wurgler, and Y, Yuan 2012 use a smaller set of proxies to construct sentiment indices for Canada, France, Germany, Japan, Uk and US, However, their proxies and indices are only available on a yearly basis. Annual data would require a very long time horizon to empirically test the performance of an investment strategy rigorously,^{12} Luckily, monthly data of a joint set of proxies from Baker and Wurgler 2006 and Baker, Wurgler, and Y, Yuan 2012 could be obtained for the three markets Europe (EU), Japan (JP) and United Kingdom (UK), These markets correspond to the three major stock market indices EUEOSTOXX50 (EU), NIKKEI225 (JP) and FTSE100 (UK).^{13}

Since the BWUSSI is well-established as well as repeatedly updated and cheeked, it is reasonable to use this sentiment index for the S&P500 (US) rather than computing one on his own,^{14} For EU, JP and UK, the indices are computed on a monthly basis from the joint set of proxies in Baker and Wurgler 2006 and Baker, Wurgler, and Y, Yuan 2012, Section 2,5 shows that the stock market indices are strongly correlated. Therefore, a portfolio consisting solely of very few stock market indices has only small potential for significant gains when incorporating sentiment into the optimization process. Apparently, such a long-only portfolio would give no opportunity to exit the stock market when sentiment would globally advise to,^{15} That is why the US Dollar Gold Price per troy ounce (GLD) is added as an alternative risky asset to the portfolio, GLD is a good choice because it often performs conversely to the stock market. If global sentiment can accurately predict negative stock market returns, then GLD should receive higher weight in the portfolio for the respective period. For that purpose, a global sentiment index is computed from the individual stock market sentiment indices as in Baker, Wurgler, and Y, Yuan 2012 (see Section 2,2),

The first sentiment proxy in Baker and Wurgler 2006 is the elosed-end fund discount (CEFD), i.e, the discount (or premium) if the elosed-end fund is trading below (above) its net asset value, A high discount should arise when investors are sceptical about the future performance of the fund. Thus, a negative relationship between the elosed-end fund discount and sentiment is expected. The discount is calculated by considering all listed equity funds on “Morningstar Direct”, Thereby, each elosed-end equity fund is assigned to one market according to Morningstar’s global category. After removing outliers, the net- asset-value-weighted average elosed-end fund discount in each market is computed,^{16} The best timing for an initial public offering (IPO) is when investors are bullish and therefore high returns by the IPO are expected (Baker and Wurgler 2006), This is the theoretical background for the number of IPOs (NIPO) being a good proxy for investor sentiment, SDC Platinum, provides the number of IPOs by market and month. As in Baker and Wur- gier 2006, the sum of IPOs over the last twelve months is used to smooth the data.

The third measure is log market turnover (TV), Following the definition of Datastream, TV is the ratio of turnover volume of all stocks listed in the market in the current month divided by the market’s value of the previous month,^{17} High TV is often an indicator for a “bubble” forming in the market and should therefore be positively related to investor sentiment (Baker and Wurgler 2006),

The last sentiment proxy is the volatility premium (VP), Baker, Wurgler, and Y, Yuan 2012 define it as the log difference of the value-weighted average market-to-book ratio MTBR between the stocks with the 30% highest and 30% lowest beta-adjusted idiosyncratic volatility σ. Beta-adjusted idiosyncratic volatility is the volatility that is asset-specific, i.e, it is not explained by movements in the common factor that affects the whole stock market. It can be computed by first estimating the following linear model for each market m

Abbildung in dieser Leseprobe nicht enthalten

where Гг is the return of asset % in market m, a is some constant (the regression intercept), rm is the return of market m, ßi captures the sensitivity of the asset return to the market return, and ti is the asset-specific (idiosyncratic) error term. The R^{2} of this regression measures the proportion of the variance in the asset returns that is explained by the variance in the market returns. Conversely, the regression’s residual sum of squares measures the proportion of the variance that is not explained by market factors. This is the idiosyncratic volatility and it is computed for each asset in the constituent list of the respective stock market index. Based on idiosyncratic volatilities, the volatility premium at time t is then formally defined as

Abbildung in dieser Leseprobe nicht enthalten

where

1ш = {г:аг> Fß^{1} {0.7)} Vt e m e {EU, JP, UK} (2.3)

Table 2.1: Summary of sentiment

Abbildung in dieser Leseprobe nicht enthalten

is the set of high-volatility stocks in market m and

Abbildung in dieser Leseprobe nicht enthalten

is the set of low-volatility stocks in market m, with Cit (Cjt) being the market capitalization of stock % (j) in period t.s

Baker, Wurgler, and Y. Yuan 2012 point out that the volatility premium is related to the dividend premium considered in Baker and Wurgler 2006. In contrast to the dividend premium, it can be computed monthly for all markets over a long time horizon. As Baker, Wurgler, and Y. Yuan 2012 illustrate both on a theoretical and empirical basis, stocks that are hard to value attract noise traders who are willing to buy these stocks even at high prices. Hence, a high volatility premium should indicate a strong sentiment effect on these stocks.

Table 2.1 summarizes the set of sentiment proxies.

Abbildung in dieser Leseprobe nicht enthalten

### 2.2 Sentiment Index Computation

As in Baker and Wurgler 2006, monthly sentiment indices are constructed from the proxies in Table 2.1 using principal component analysis (PCA). The procedure is based on the intuition that all variables are driven by one latent factor: investor sentiment. To cheek whether PCA is applicable, Bartlett’s sphericity test (Bartlett 1951) is conducted. It tests the hypothesis Ho that a correlation matrix R is equal to the identity matrix I, i.e.

Abbildung in dieser Leseprobe nicht enthalten^{18}

where

Abbildung in dieser Leseprobe nicht enthalten

with rij being the Pearson correlation between two variables Xi and Xj.

The test statistic to compute is

Abbildung in dieser Leseprobe nicht enthalten

where n is the number of observations, p is equal to the dimension of R, and \R\ is the determinant of the correlation matrix. Under H0, the test statistic follows a x^{2} -distribution with V = p(-p^{1} -> degrees of freedom.

P-values of the test are always close to 0 so that Ho is strongly rejected for each market’s proxies and PCA is applicable.

Baker and Wurgler 2006 define their BWUSSI as the first principal component (PC) of their sentiment proxies. The first PC is a linear combination of the proxies that accounts for as much joint variation of the proxies as possible. Mathematically, it is a variable y which is a linear combination of some variables X with the highest variance

Abbildung in dieser Leseprobe nicht enthalten

where w are the weights (loadings on the PC) for X .

Let s be an estimator of the covariance matrix Σ, the optimization problem becomes

Abbildung in dieser Leseprobe nicht enthalten

The constraint requires that the weights add up to 1. The variance is maximized for

Abbildung in dieser Leseprobe nicht enthalten

where Λ is the largest eigenvalue of matrix s and w is the corresponding eigenvector.

For the dataset used in this study, the first PC explains respectively 39% (EU), 40% (JP) and 44% (UK) of the proxies’ variance. These proportions are similar to those found in Baker and Wurgler 2006 and Baker, Wurgler, and Y. Yuan 2012. The investor sentiment indices (ISI) are computed as

Abbildung in dieser Leseprobe nicht enthalten

With the exception of the JP TV and UK CEFD all loadings have the expected sign.^{19} A technical note is worth mentioning here. Since the PC maximizes the joint variance, the signs of the loadings are exchangeable. In fact, for the EU and UK the indices are mirrored because after exchanging the signs, the loadings are in sum more intuitive. Furthermore, the mirrored indices exhibit higher correlation with the BWUSSI. Even though not perfectly correlated, market sentiments most likely share some patterns. Thus, the other indices should not differ too greatly from the well-established BWUSSI. More pre- eisely, pronounced stock market events like the global financial crisis of 2008 or the dot com bubble between 2000 and 2002 are clearly observable in the BWUSSI. Due to the global reach of the crisis, the corresponding sentiment peaks and troughs should therefore also be observable in the other developed market sentiment indices.^{20}

The regional investor sentiment indices are standardized as in Baker and Wurgler 2006. Then, in analogy to Baker, Wurgler, and Y. Yuan 2012, a global sentiment index is computed as the standardized first principal component of the regional sentiment indices for the EU, JP, UK and the US (BWUSSI). The global index reflects global investor sentiment towards the stock market and is therefore inversely related towards the alternative asset GLD. It is computed as

Abbildung in dieser Leseprobe nicht enthalten

Apparently, all regional sentiment indices contribute about equally to global sentiment. This is in contrast to Baker, Wurgler, and Y. Yuan 2012 who find that US sentiment has the strongest impact on global sentiment.

Figure 2.1 depicts the regional sentiment indices and the global sentiment index over time. They are in accordance with major stock market episodes. For example, all indices reach a very high level of investor sentiment prior to the global financial crisis starting in 2007 and experience a large drop as the crisis unfolds. Also the recovery from the crisis starting around 2010 is visible. Likewise, the period of the dot com bubble is captured by sudden shifts in investor sentiment before and after the bubble’s collapse between 2000 and 2002, with the exception of the JP index. Unsurprisingly, the global crises and recoveries are also observable in the global sentiment index,

Huang et al, 2015 suggests an alternative partial least squares (PLS) approach to compute the BWUSSI in a way that it is more aligned to market returns. They find that the sentiment aligned index is a better predictor of the stock market. In the Appendix, i, it is shown how to compute the aligned sentiment indices for the EU, JP and UK, Figure A.l depicts the sentiment aligned indices over time and shows that they are more noisy and, from an eyeball test, much less in accordance with major stock market episodes as the PC sentiment indices. Therefore, the PC indices are preferred in this study,^{21}

The following section describes the Blaek-Litterman (BL) and Copula Opinion Poooling (COP) methodology. The latter is used to incorporate the investor sentiment indices into portfolio optimization.

Figure 2.1: PC sentiment indiees by market over time (a) EU (b) JP

Abbildung in dieser Leseprobe nicht enthalten

Each index represents the standardized first PC of the sentiment proxies. For the US, the sentiment index by Baker and Wurgler 2006 is used. The global sentiment index is derived as the first PC of the regional sentiment indiees and inversely determines the portfolio weight of GLD.

### 2.3 Incorporating Sentiment into the Portfolio

#### 2.3.1 Black-Litterman Approach

Black and Litterman 1992 motivate their paper by showing that previous standard optimization techniques lead to extreme portfolio weights - or corner solutions when setting constraints - that are often undesired by investors who only like to differ slightly from the capitalization-weighted market portfolio. The reason is that classical optimization techniques strongly depend on the expected return and volatility that are often estimated from historical data and not necessarily good predictors of future performance. Black and Lit- termán 1992 solve this shortcoming by providing a tool for investors to blend their market views with the market prior and retrieve a well-behaved portfolio in a mathematical sound way.

The basic idea is to first establish a market neutral portfolio that serves the investor as a reference point. Thereby, neutrality follows a market equilibrium argument. If all investors had the same view about expected returns and wanted to hold the same portfolio, then supply and demand would not be in equilibrium and market prices would have to adjust. Thus, neutrality refers to the expected returns where supply equals demand. The corresponding distribution is referred to as the market prior distribution and prior market weights are determined by relative market capitalization.

Abbildung in dieser Leseprobe nicht enthalten

Second, the market posterior distribution is estimated conditional on the investor’s views about the market. Intuitively, the estimation approach can be understood with Bayesian probability theory. Using Bayes’ theorem

where M would represent the market portfolio and V would represent the investor’s views.

The following more formal description of the Black-Litterman approach is based on Black and Litterman 1992 with great help from G. He and Litterman 1999 and particularly Walters 2014. For ease of understanding, Walters 2014 uses the well-known Capital Asset Pricing Model (CAPM)^{22} together with a quadratic utility function to demonstrate the computation of equilibrium market returns under mean-variance optimization.

Let there be n assets and assume that their excess returns follow a multivariate normal distribution

Abbildung in dieser Leseprobe nicht enthalten

where r is the asset return vector, г ƒ is the risk-free rate, μ is the vector of expected excess returns, and Σ is the covariance matrix of the returns,^{23} The expected return of asset % is again a random variable that, under CAPM, can be estimated using a linear regression model of the form

Abbildung in dieser Leseprobe nicht enthalten

The parameter ßi = measures the sensitivity of the returns of asset % to the returns of the market rm, E(rm) is the expected market return, and ti is the residual or asset-specific idiosyncratic return,^{24} In the next step, the CAPM equation is simplified to

Abbildung in dieser Leseprobe nicht enthalten

where π is the vector of equilibrium excess returns. Under the normality assumption of the linear regression model, e is a vector of i.i.d, random variables, i.e, e ~ N(ο, Σπ), Thus, μ is centered around the equilibrium excess returns and one can write down the distribution of the mean equilibrium excess return as

Abbildung in dieser Leseprobe nicht enthalten

where π is the estimate of the mean with unknown variance Σ-π-, Intuitively, the uncerlai ni у about the mean will be less than the uncertainty about the individual returns so

that Σ7Γ < Σ,

Market Prior Distribution

Within the CAPM, all investors hold the same market portfolio that, under mean-variance optimization, is the market portfolio with the highest Sharpe Ratio (see Section 2,4), However, investors cannot all demand the same risky portfolio as long as the market is

Abbildung in dieser Leseprobe nicht enthalten

not in equilibrium. At equilibrium, investors hold a portfolio with weights Wi equal to the relative market capitalizations of the assets

where Ci is the market capitalization of asset i.

Abbildung in dieser Leseprobe nicht enthalten

To find the corresponding equilibrium excess returns, the optimization process is “reversed”. Generally, the objective is to maximize the utility of the investors, Walters 2014 uses the quadratic utility function

where A is a constant scaling parameter that measures the average risk tolerance of the investors,^{25} Utility is maximized by taking the first order derivative of и with respect to the weights w

Abbildung in dieser Leseprobe nicht enthalten

The weights that clear the market are known and given in Equation (2,23). Reversed optimization means to solve the equation for the implicit equilibrium excess returns. The equilibrium excess returns are

Abbildung in dieser Leseprobe nicht enthalten

But π cannot be computed without having an estimate of A, Multiplying both sides of Equation (2,26) by W׳ and solving for A, the following relationship surfaces

Abbildung in dieser Leseprobe nicht enthalten

where E{rm) is the expected market return, am is the market volatility and SRm is the Sharpe Ratio of the market. Thus, one way to specify the risk aversion parameter A is to come up with an estimate of the Sharpe Ratio of the market. This estimate is often based on expertise or calibrated from historical data (Walters 2014, p, 10),

After calibrating A, taking Σ as given (e.g, also by estimation from historical data), and using the market cap weights from Equation (2,23), the equilibrium excess returns π are

**[...]**

^{1} See Section 2.2 and Section 2.5 for why GLD is added to the portfolio.

^{2} Mostly however, the effects related to the stock market are investigated, and there again, most studies

are tied to the US market.

^{3} However, Brown and Cliff 2005, p. 414-415, note that any effect is subject to the joint hypothesis problem, which results from the possibility that the pricing model by Bakshi and z. Chen 2005 is misspecified.

^{4} For example, Pàstor and Stambaugh 2003 find that stocks with higher sensitivity to mar ket-wide liquidity have average higher expected returns (liquidity factor). Jegadeesh and Titman 1993 discover that past winners perform better than past losers (momentum factor). The search for new factors has become so extensive that data mining is considered an issue. That is why Harvey, Liu, and Zhu 2016 suggest to change the usual testing procedure to adjust statistical significance so that the hurdle is higher for more recent claims of new pricing factors.

^{5} Baker and Wurgler 2007, p. 135-138, provide a more extensive list of other possible sentiment measures.

^{6} Section 2.2 elaborates more on the construction of sentiment indices using PCA.

^{7} Abbreviated are the proxies used in the empirical study of this thesis.

^{8} E.g. momentum, financial distress, total accruals, etc. See Stambaugh, Yu, and Y. Yuan 2012 for the complete list and explanations of the anomalies.

^{9} The proxies used in the empirical study of this thesis are abbreviated.

^{10} Of the literature reviewed in Section 1.1, for example, Schmeling 2007, Heiden, Klein, and Zwergel 2013 or Sul, Dennis, and L. Yuan 2017 successfully implement a trading strategy based on their empirical results.

^{11} Notable exceptions in Section 1.1 include Schmeling 2007, Heiden, Klein, and Zwergel 2013 or Sul, Dennis, and L. Yuan 2017.

^{12} Say the time horizon is 20 years, then there would be only 20 portfolio decisions based on the sentiment data. One can hardly argue to have found a superior investment strategy from just 20 observations, which could as well be pure chance. Furthermore, if - for robustness or out-of-sample studies - the data is split into a training and test period, this issue becomes even more pressing.

^{13} From now on, it is referred to the respective index when speaking of the abbreviated market in brackets.

^{14} The BWUSSI can be downloaded from Jeffrey Wurgler’s website http://people.stern.nyu.edu/ jwurgler/.

^{15} For more details about this, see also Section 2.4.

^{16} Outliers are all discounts, CEFD0UT, that are in the lower and upper 1% empirical quantile, i.e. CEFD0UT < TjļFD(0.01) and CEFD0UT > Fj1fd(0.99).

^{17} In contrast to Baker, Wurgler, and Y. Yuan 2012, TV is not detrended because there is no visible positive trend in the dataset of this study.

^{18} Negative MTBRs are removed as otherwise this leads to negative arguments in the log ratio in some months.

^{19} Note that the loadings do not add up to 1 because the function to conduct principal component analysis in R (prcomp) uses a different optimization procedure called singular value decomposition which achieves better accuracy.

^{20} However, this way of calibrating the sentiment indices should be rethought in case the markets are more regarded as contrarily by investors, e.g. emerging vs. developed markets.

^{21} Not unexpectedly, the results using the PLS indices are also worse compared to using the PC indices. To conserve space, the results using the sentiment aligned indices are not reported.

^{22} The CAPM was independently developed by Lintner 1975; Mossin 1966; Sharpe 1964; Treynor 1961a; Treynor 1961b. Note that an edited version of Treynor 1961b is available in the book Treynor 2012. The CAPM builds on Henry Markowitz’s Modern Portfolio Theory which is explained in detail in Section 2.4.

^{23} Note that any covariance matrix is independent of a linear transformation, here subtracting Tƒ from asset returns r, so that Σ is the covariance matrix of both the returns and excess returns. However, this is only true as long as Tf is the same for all assets. For example, if there are assets from different countries, the excess returns are empirically computed from different risk-free rates (the different government bond yields).

^{24} CAPM postulates a linear relationship between risk and return. Thereby, ßi measures systematic risk that can be diversified by selecting assets that are low or even negatively correlated with the market (pi,mOn the other hand, the lower ßi, the less the investor participates in bullish markets. Unsystematic risk, i.e. risk that cannot be diversified since it is asset-specific, is measured by Cļ. Note that in Black and Litterman 1992, the impact of the common factor z, 7i, is analogous to the CAPM ßi, and the asset-specific shock component Vi is analogous to the CAPM idiosyncratic Cļ.

^{25} More specifically, it determines the proportion of expected return investors are willing to give up for less risk (Idzorek 2007, p. 3).

- Quote paper
- Nicolas Banholzer (Author), 2018, Exploiting Investor Sentiment for Portfolio Optimization, Munich, GRIN Verlag, https://www.grin.com/document/425070

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