Stock market volatility: A puzzle? An investigation into the causes and consequences of asymmetric volatility

he noted that noise makes financial markets possible, but also makes them imperfect. In addition to this, he pointed out that people who trade on noise are willing to trade even though, from an objective point of view, they would be better off not trading. Perhaps they think the noise they are trading on is information. In doing this continuous trading without relevant firm news, Black supposed that this would cause the large fluctuation of the stock market. Hence noise trading combined with “financial leverage” is the main volatility driver of the stock market. The two effects combined could also perhaps affect the asymmetric volatility found in empirical research. We already know the low level of empirical support for the leverage effect and, from the literature, we also know that the noise trader models are not fully convincing. De Long, et al. (1990) follow a somewhat different direction of reasoning about noise traders. In fact, they postulated that many investors do not follow the economic advice of market efficiency. Therefore, they do not buy the market portfolio, instead taking a position on single stocks following some rather “irrational” chart based trading strategy. In reality, we know that if markets are efficient, then technical analysis is useless in making a decision to buy or sell stocks. Their main point is that arbitrageurs are risk averse and have reasonably short investment horizons. As a consequence, the positions they are willing to hold against noise traders are limited. Stock Markets Volatility: A Puzzle? Boris F. Papa - 61 - Noise traders falsely believe that they have special information about future prices of the risky assets. Thus, they select their portfolios on the basis of such incorrect beliefs. But there is also a second source of risk, the risk that noise trader activity is strong enough to stop prices from the mean reverting over a long time. If these noise traders are able to move prices away from the fundamental value, than rational arbitrageurs, who select their trade based on fundamental information, could suffer losses because of this noise trading. As a consequence, they will not bet against the noise trader decision, although they have better information and they know the right price. So in this battle, the noise traders move asset price sharply away from the “fair value”. We ask ourselves, how is it possible that only the arbitrageur would suffer from losses and not also the noise trader, who, in the end, will just obtain gains for luck, yet on the average and in the long run, should not earn “free money”? Connected to this is the still-open question of whether the arbitrage is limited. The answer is not clear- cut. But evidence that two different shares of the same company (such as Shell and Royal Dutch) could be traded at different prices for a long time, is some evidence for persistence in mispricing, and the consequences for trying to exploit this mispricing could be fatal (see the experience of LTCM). De Long, et al. (1990) reasoning is different. They conclude that noise traders are rewarded for bearing the risk they themselves create. As a result, noise traders will earn higher returns from holdings these assets. But this conclusion somewhat contradicts standard finance theory because it postulates that only undiversifiable risk will have a premium, since this cannot be taken away. Hence, the diversifiable risk should not bear any risk premium. It seems to us that this noise trading is rather a diversifiable risk. De Long, et al. developed an analytical model, which we cannot examine here for reasons of time and space. They applied these models to some market anomalies, such as the closed-end-puzzle, the equity premium puzzle and others. With this model, it should be possible also to explain the excess volatility because noise traders push price far from the fundamental one and, in doing so, the risk, i.e., volatility, may rise. So in the end, the argument is just slightly more formal than that of Black (1986), but not sufficiently so to explain the observed market volatility. We are also sceptical of the argument because it seems nearly impossible that this category of traders, which seems to be very small in practice, could have so a large an impact on asset price for as big a stock market as the USA. For this reason, we do not follow this theory any further in explaining stock market volatility and, instead, go to one (perhaps the only one at the moment) asset pricing theory, which makes use in some form of the behavioural finance and prospect theory, in particular, to explain volatility. 4.1.3. Prospect Theory and Asset Pricing We will describe in this section the model derived by Barberis, et al. (2001) in a very influential paper. This is (at least to our knowledge) the first attempt to implement the new findings of prospect theory and behavioural finance in an asset pricing model. Moreover, it is perhaps the first attempt to combine the “classical framework” of maximisation of final consumption over an infinite time horizon, with the prospect theory findings of loss aversion. Therefore, the model presented here is not only interesting for theoretical reasons, but could have a very large impact on asset pricing theory. Barberis , et al. (2001), began by describing three “puzzles” that the classical standard finance theory of consumption-based model cannot explain. The three identified puzzles are: 1) the equity premium puzzle; 2) the volatility puzzle; and 3) the predictability puzzle. We have already seen and described the first one. They defined the volatility puzzle as the stock market levels that appear to move around too much (standard excess volatility). The argument here is as follows. Price to earnings has been historically very high (at least for the USA market), which can be interpreted as future high cash flows of firms. But historical data show that higher earnings do not follow historically higher P/E ratios. The puzzle exists in the question of why the price was so high in the beginning, i.e., the volatility puzzle. The predictability puzzle goes in the opposite direction: There is a lot of empirical evidence that high P/E ratios are followed by low returns and vice versa. Stock Markets Volatility: A Puzzle? Boris F. Papa - 62 - 4.1.3.1. Investor preferences and the main assumptions The starting point of the model is the traditional classical model of Lucas (1978), that we have described extensively in Chapter 3. We use the same notations as the original paper for at least two reasons. First of all, so it will be consistent with the original one, and, secondly, to make a clear distinction between this model and the many we have seen in chapter three, although the methodology will remain more or less the same. The model works with the classical standard power utility function for the consumption part and a modified version of the cumulative prospect theory of Tversky and Kahneman. In particular, agents are assumed to maximise following expected value: ( )                 +−Ε ∑ ∞ = + + − 0 2 1 1 1 1 1t ttt t t tt zSXvbC      ,,ργρ γ (4.7) The first term (1) is the standard classical maximisation problem that we have seen in chapter 3 (see above for interpretation of the variables). The C is just the consumption, the factor ρ is the standard time discount factor for time t or for time t+1, and γ is the common curvature of utility over consumption. The second term (2) represents utility from fluctuations in the value of financial wealth and incorporate an extended prospect theory “value function”. This is a function of the gains or losses ( 1+tX ) over the next period, the security price ( tS ), and a state variable ( tz ) which measures the investor’s gains or losses prior to time t as a fraction of tS ; in other words, the past performance of risky asset. Finally, tb is an exogenous scaling factor. From (4.7), one can see why we called this new approach to asset pricing “the modern finance synthesis” because it combines the classical consumption theory and the behavioural part in which investors derive utility also in the future stock market performance. Nothing is said in this model about rationality of investors, but the authors pointed out that this model is not incompatible with rationality and, in this sense, is just an extension of the classical model that we have already seen with the Epstein and Zin utility function. To follow this, we should briefly overview the variables, which enter into the model. Gains and losses ( 1+tX ) are measured with respect to the returns on the risky asset, but also scaled up by the risk free rate to find the reference level. This means that investors are interested in the excess return over some period (1 year for practicality) and derive from this utility: tftttt RSRSX ,−= ++ 11 where tS is the stock price, 1+tR is the next period return and tfR , is the risk free return over the next period know in advance at time t. To capture the influence of prior outcomes, they introduce the concept of historical benchmark level tZ for the value of the risky asset. This is just to capture the fact that a loss following another loss has a much higher impact on an investor’s risky asset holdings than a loss after a long period of gains. It introduces a dependence structure on the past performance quite similar to the reasoning that we have seen in habit formation of Campbell and Cochrane (1999). In contradiction to this model, this time the reference is not the consumption, but the portfolio returns. For modelling purposes, they find it convenient to write the scaled past performance as t t t S Zz = . From this, it follows that we can distinguish three distinct cases: 1=tz , where the investor has neither prior gains nor prior losses on Stock Markets Volatility: A Puzzle? Boris F. Papa - 63 - his investments; 1tz , the case of prior losses. For each of these cases, they derived a functional form of the utility function. Starting with 1=tz , or just to catch the loss aversion of investors, we have ( ) 0 0 1 1 1 1 < ≥  = + + + + + 1t 1t t t tt X X for X X SXv λ,, (4.8) where λ>1 is. The intuition here is that an investor with a cushion of past gains may not care about small losses, but large losses may matter. For the case of 1≤tz , considering a scaling risk free rate, we have the following functional form: ( ) ( ) ( ) tft1t tft1ttfttttftftt tftttttt RzR RzR for RzRSRRzS RSRS zSXv , , ,,, ,,, < ≥   −+− −= + + + + + 1 1 1 λ (4.9) Finally, we have to consider the case of 1>tz , when investors have experienced some losses, we have: ( ) ( ) 0 0 1 1 1 < ≥  = + + + + + 1t 1t tt t ttt X X for Xz X zSXv λ,, (4.10) where ( )tzλ >λ is, and this considers some penalty function dependent on past losses: ( ) ( )1−+= tt zkz λλ (4.11) where k>0 is. The larger the past loss, the more painful subsequent loss will be. These three characterizations fully describe the utility function of the investor derived from the stock market and past returns. If we look more closely, we can see that the functional form of (4.8) is somewhat similar to the original prospect theory proposed by T&K. However, the other two are clearly new extensions,which captures the fact that past loss may influence more or less the investor reaction to a new downward market, dependent upon whether the past has been very successful or less successful. These three characterizations determine the curvature of the value function in the negative part: it is more pronounced in the third case and less in the first one, although the curvature changes suddenly if the investor perceives the loss could be persistent and so becomes more loss averse. Finally, we should consider the scaling factor tb , which is important to maintain the stationarity of risk premium because an increasing second term would imply a domination over the first as wealth increases. To satisfy this condition, the authors introduce the scaling factor as: γ−= tt Cbb 0 (4.12) where tC is the aggregate per-capita consumption at time t, and hence exogenously given. 0b is a positive constant factor. If it is zero, then (4.7) it reduces just to the traditional standard classical model. Thus, the proposed model of Barberis, et al., incorporates some features of prospect theory, but not all. They neglect the transformation of probability, for instance, and the value function is a little bit different, as we have seen above. But the most important fact that investors are loss averse, if they have already had a loss, is well implemented. On this point, there is some contradiction in the literature because some authors consider “risk seeking” behaviour of investors after a loss. This is Stock Markets Volatility: A Puzzle? Boris F. Papa - 64 - experimentally unfounded, and also, in practice, people consider loss averse investors as the “normal case”. Of course, if an investor has a lot of losses and perceives for any reason some probability of rebound of the market, than he becomes more risk seeking. This feature can very well be observed in other games like casino, slot machine and poker. People that lose a lot try everything to a least break even. The literature on probability is rich here: The ruin theory is commonly known as “don’t play the martingale!” In this case, if you play the supermartingale, you will, on the average lose money almost surely. 4.1.3.2. The model In the original (Barberis, et al. 2001) paper, the authors derived the asset pricing formula in two different economies. The first is like the Lucas model, where the dividends and consumption follow the same process. The second is where the dividends and consumption follow two different processes. We will have a look only at the second because the first economy is too similar to the models we have seen in chapter 3. In addition to this, the second is so general that it includes the first as a special case. They assume that the risk free rate is constant and the equilibrium is a one-factor Markov, driven by the state variable tz , which determines the distribution of future stock returns. In addition to this, the price-dividend ratio of stock is a function of the state variable tz : ( )t t t t zfD Pf =≡ (4.13) Given the one-factor assumption, the distribution of stock returns 1+tR is determined by tz and the function f() using: ( ) ( ) t t t t t tt t D D zf zf P DPR 1111 1 +++ + +=+= (4.14) Than the processes followed by consumption and dividends are: 1 1 + + +=    tCC t t g C Clog ησ (4.15) 1 1 + + +=    tDD t t g D D log εσ (4.16) where                 1 1 0 0 w w Ndii t t ;...~ε η This last assumption, which makes     t t D Clog a random walk, allows us to construct a one-factor Markov equilibrium in which risk free rate is constant. This means that the stock returns is ( ) ( ) 111 1 ++++ += tDDg t t t ezf zfR εσ (4.16) Stock Markets Volatility: A Puzzle? Boris F. Papa - 65 - The first order conditions for an optimum are than a pair of Euler equation:        Ε= − + γ ρ t t tf C CR 11 , and (4.17) ( )[ ]ttt t t tt zRvbC CR ,ˆ 10 1 11 + − + + Ε+       Ε= ρρ γ (4.18) where for 1<tz , ( ) ( ) ( ) tft1t tft1ttftttftft tfttt RzR RzR for RzRRRz RR zRv , , ,,, ,,ˆ < ≥   −+− −= + + + + + 1 1 1 λ (4.19) and for 1>tz ( ) ( )( ) tf1t tf1ttftt tfttt RR RR for RRz RR zRv , , , ,,ˆ < ≥   − −= + + + + + 1 1 1 λ (4.20) This optimal equation are similar to those seen in chapter 3, also the interpretation is similar. The only changes here is that consuming less today to invest in the risky asset exposes the investor to greater losses, which depends on the state variable tz . Given this they has been able to proof the following proposition: Proposition 4. 1: The risk free is constant at 21 22 CCg f eR σγγρ −−= (4.21) And the stock’s price-dividend ratio f() is given by: ( ) ( ) ( ) ( ) ( ) ( )        +Ε+   +Ε= ++ ++−+−+− tg t t t w t t t wgg ze zf zf vbe zf zf e tDDtCDCCD ,ˆ 11 222 1 0 112 1 11 1 εσεσγσ σγγ ρρ (4.22) where vˆ is given by (4.19) and (4.20). Proof: See Barberis, et al. (2001, Appendix) 4.1.3.3. Model results and interpretation If you take the logarithm of formula 4.21, the risk free rate is similar to that derived in (3.50) above, apart from some other parameters. The second equation in (4.22) is really complex and not so easy to understand. Moreover, there exists no closed form analytical solution, and it is necessary to run numerical methods to solve it. The authors pointed out that in numerical tests, this model has been able to show low consumption volatility and high stock returns volatility, while maintaining low stable risk free rate and low correlation of stock returns to the consumption. These results, of course, comply well with the findings in many empirical studies. Thus, one may conclude that this model is an improvement of the habit-formation one of Campbell and Cochrane, and can well explain the asset pricing and the numerous puzzles found with classical models. In addition to this, it is easy to explain the high equity premium and also the asymmetrical volatility. Investors are loss averse; therefore they ask high risk premiums to prevent them from losses in holding risky assets. Also, and for perhaps the same reason, they increase the volatility of stock prices when they drop. This is because they become Stock Markets Volatility: A Puzzle? Boris F. Papa - 66 - very loss averse and prefer to sell risky assets, reinvest the proceeds in the risk free rate, thus pushing down the latter and the risky asset price, pushing up the volatility of stock returns. Although the authors concluded that a model that relies on loss aversion alone cannot provide a complete description of aggregate stock market behaviour, they added that the fundamental weakness of such a model comes in explaining the volatility which is too low to be compatible with the real data. We can conclude by saying that this is a great step forward in explaining asset pricing and loss aversion, but perhaps it needs some degree of refinement. We can also comment that this is “reversal engineering” modelling, as in Campbell and Cochrane (1999). These models are constructed so that they can provide a good explanation of the data, but many unanswered questions remain about prediction and reliability of such models. In our opinion, there are still too many assumptions. The models are very sensitive to the estimate of the parameters, which can be seen as the major drawback. Whether or not these models are in compliance with other models in macroeconomics that explain an agent’s decision-making and behaviour is still an open question. Also, the attempt to combine the classical with the new prospect theory is courageous on the one hand, but is misleading on the other because of the potential difficulty in really supporting the validity of the two theories combined. Nevertheless, future research will probably involve a further attempt to combine the better of the two theories. More psychology must be included in these models, but how do we model human behaviour and human psychology with a simple mathematical model? If we cannot answer this question, then models may be of little or no use. 4.2. Conclusion from Non-Classical Finance “I don’t believe in mathematics” Albert Einstein Behavioural finance is a new branch of financial economics that attempts to analyse and explain many financial market anomalies using non-standard financial methods. The market efficiency hypothesis, the rationality of investors, and above all, the expected utility function failed completely to explain market returns and agent decision-making. From this starting point, behavioural finance raises many human preference and belief hypotheses and, using psychological phenomena, it also tries to explain the numerous puzzles. The most important of the mainstream behavioural theories is the Prospect Theory founded by Kahneman and Tversky in 1979. They further developed it in 1992, leading to the cumulative prospect theory. The central idea of this new theory is the fact that people overestimate lower probabilities and underestimate higher ones. In addition to this, people are risk averse for gains and risk seeking for losses. This leads to an S-shaped value function. The main idea is that investors are loss averse, trying to do everything to avoid losses. Barberis, et al. (2001) was one of the first to use the prospect theory value function in connection with some standard consumption power function. The idea has been to integrate into one model the utility derived from consumption with that derived from investment in the risky asset. These authors introduced a new feature of loss aversion. Investors care about past performance in the stock market. They begin to change their attitude toward risky assets if they have registered successive losses. Within this complex model, where some parameters are questionable, one is in the position to explain many asset pricing puzzles. In addition to this, through the loss aversion, investors will cut losses, thus suddenly inducing the asymmetric volatility phenomena. The authors provided many numerical simulations to show the validity of such modelling. In addition to the well-known problem of consumer-based models, this new feature of past performance dependency of the value function makes the testing and tractability of such a model very difficult. Nevertheless, we are convinced that the breakthrough of this new model could lead to some interesting further research and results. Stock Markets Volatility: A Puzzle? Boris F. Papa - 67 - Behavioural Finance has had much criticism from “classical” finance researchers, foremost Eugene Fama, who believes in the efficient markets hypothesis. Another well-known researcher is Mark Rubistein (2000, p.17), who maintained that “many so called anomalies are empirical illusions created by data mining, survivorship bias, selection bias, short-shot bias, trading costs, and the high variance of sample means”. We will not comment here on this statement. However, we should at least admit that empirical research is not the last word because, depending upon how one selects data, selects the data time period, and looks at or analyses the data, one can find some is used, interesting relationships or contradictions of some other research. The best example is the momentum strategy of Jegadeesh and Titman (1993). In fact, their empirical findings show how past winners consistently outperformed past losers, if one uses a window of six to twelve months. Contrary to this, De Bondt and Thaler (1985) find empirically that up to a five-year portfolio of past losers beat a portfolio of past winners. They explain this with representativeness heuristic. Can this test contradict the former one? Or is it just another test? Many of these questions related to empirical finance are still unanswered. Nonetheless, we can claim that behavioural finance has revolutionized financial economic theory and changed it for the better. Future research will show us if it will remain a separate branch of financial economics or if it will become an integrated part of standard theory. We are convinced that future research will be more important. One cannot neglect human behaviour and anomalies in analysing risky asset pricing and volatility. In the end, stock prices, returns and volatility are always and everywhere the results of human action and not always in compliance with models or quantitative financial analysis. Stock Markets Volatility: A Puzzle? Boris F. Papa - 68 - 5. Conclusion “Man is not a circle with a single centre; he is an ellipse with two foci. Facts are one, ideas are the other” Victor Hugo, Les Miserables In this paper we have analysed international stock market volatilities and found the commonly- known stylised facts. In addition to this, one of the characteristics of stock market volatility is the asymmetric behaviour: When the stock market declines, volatility increases more than when the stock market rises. This asymmetrical behaviour of stock market volatility represents a puzzle that standard financial economics cannot explain. To some extent, new consumption-based models that include habit formation can explain this puzzle. However, if one looks in detail, this is the result of some financial engineering, which can be difficult to explain with economics. The habit formation model of Campbell and Cochrane (1999) is one of these successful examples. They provide at least some support for the empirical validity of such a model. Within this model is also a possible explanation of some stylised facts about consumption. In fact, the consumption smoothing and low correlation with the stock market can be explained well. Asymmetrical volatility is the cause of skewness and fat tail in the distribution of returns. This leads to many problems for the standard model in finance. For the mean-variance approach, it means that the variance is very high in a downmarket when the returns are very low, leading to inefficient portfolios. This can be explained by a trader view of “rebound.” For the CAPM, asymmetrical volatility means higher correlation, which should compensate the higher variance, but also lower expected returns; or, if we do not assume it as a constant, higher equity premium. This is not the case in empirical findings. The consequences for the Hansen-Jagannathan lower bound is that the assumed complete market and perfect correlation no longer work. As a result, the stochastic discounted factor is very volatile and the bound is not so well defined. The option pricing theory is also challenged by the stochastic volatility. The market is incomplete and there is difficulty in finding a risk neutral probability, which makes a discounted security a martingale. In addition to the volatility smile, one may find that there is higher implied volatility when the market corrects than when the market rises. The theory of long call when this happens has been exposed, although it is not very convincing. Analysing the VIX index, we find that negative return is related to higher change in implied volatility, as is the positive return. This leads to another side of the asymmetrical volatility. Corporate finance theory explains that a decline in the stock market leads to decreasing equity compared to debt and to an increase of the leverage effect, which should cause higher volatility. In many empirical studies, this leverage effect has not been very successful in explaining volatility. For example, with Enron the leverage effect can explain only a small part of the variance. The empirical evidence of anomalies and puzzles gives support for a new financial theory based on the psychological phenomenon of human beings. Prospect theory is the main framework in the behavioural finance. The primary basis of this theory is that investors are loss averse, which leads to asymmetrical behavioural to returns, thus causing the asymmetrical volatility. Application of this to asset pricing, Barberis, et al. (2001), has shown that it can explain many puzzles, including the equity premium puzzle and the asymmetric volatility. The introduction of the idea of noise trading risk raises the issue of higher volatility. These models attempt to explain how uninformed traders can limit the arbitrage and increase the volatility. Nevertheless, it remains difficult to test. Pure psychological explanation of human behaviour can also explain the asymmetric volatility. Cognitive dissonance appears when people discover that they can be wrong. To minimise this dissonance, they sell risky assets when the stock market decreases, pushing returns down and volatility up. Many classical financial economists are sceptical about the many anomalies found by behavioural finance. Fama interprets the fact that both overreaction and underreaction are observed in financial markets as evidence that the anomalies from the standpoint of efficient markets theory are just “chance Stock Markets Volatility: A Puzzle? Boris F. Papa - 69 - results.” Therefore, the theory of market efficiency survives the challenge of its critics. In fact, Fama states that the literature on testing market efficiency has no clearly stated alternative since the alternative hypothesis is vague market efficiency. Mark Rubinstein (2001) describes the “Prime Directive” of financial economists to be to: “explain asset prices by rational models, only if all attempts fail resort to irrational investor behaviour.” Steve Ross (2001) offers the following statement: “At present, behavioural finance seems more defined by what it doesn’t like about neo-classical finance than what it has to offer as an alternative”. He than added, that ”behaviouralists are surely right about one thing: most of the time most of the people do, indeed, misbehave”. Richard Thaler (1999) anticipates the future integration of behavioural finance within the “standard” paradigm: “In the future financial economists will routinely incorporate as much behaviour into their models as they observe in the real world”. If this is really so, in order to solve the puzzle, satisfactory models of „stock market volatility“ may have to wait until behavioural finance develops human behaviour models related to the stock market that can properly coordinate with standard finance economics. Only time will tell. Stock Markets Volatility: A Puzzle? Boris F. Papa - 70 - Appendices Appendix I Figure A. I: Density estimate for MSCI Stock Price Index Appendix II Stock Markets Volatility: A Puzzle? Boris F. Papa - 71 - Figure A. II: Density estimate for Swiss Stock Price Index compared to the Gaussian density Appendix III Figure A. III: Annualised excess returns and volatility of international stock markets (MSCI Price index). Stock Markets Volatility: A Puzzle? Boris F. Papa - 72 - Appendix IV Figure A. IV: Some plot of annualised GARCH volatility for the international stock markets Stock Markets Volatility: A Puzzle? Boris F. Papa - 73 - Appendix V One may consider a maximal correlation between the two SDFs and get ( )( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )[ ] ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( )22 0 VarVarVar, or ,VarVarVar, ,CVar 0 ,CVarVar ,-VarVarVar LLLLLLLLLLCorr LLLLLLLLLLCorr LLLLovLL LLLLovLLLL LLLLLLCovLLLLLLLL ttttt ttttt ttt tttt ttttttt ∗∗∗ ∗∗∗ ∗∗ ∗∗ ∗ ≈ ∗∗ = = −= −+= −++=    So, the desired result finally follows ( )( ) ( )( )( ) ( )( ) ., Var Var 2LLLLCorr LLLL tt t t ∗ ∗ = (AV.1) Stock Markets Volatility: A Puzzle? Boris F. Papa - 74 - References Abel Andrew B. (1988): “Stock Prices under Time Varying Dividend Risk: An Exact Solution in an Infinite-Horizon General Equilibrium Model” Journal of Monetary Economics, 22, 375-393. Abel Andrew B. (1990): “Asset Prices under Habit Formation and Catching Up with The Joneses” American Economic Review , 80, Papers and Proceedings, 38-42. Ait-Sahalia, Yacine, and Andrew W. 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