Essays on Asset Pricing and Downside Risk

Transcription

1 Essays on Asset Pricing and Downside Risk Bruno Cara Giovannetti Submitted in partial ulillment o the requirements or the degree o Doctor o Philosophy in the Graduate School o Arts and Sciences COLUMBIA UNIVERSITY 2

2 c 2 Bruno Cara Giovannetti All rights reserved

3 Abstract Essays on Asset Pricing and Downside Risk Bruno Cara Giovannetti This dissertation contributes to the recent and diverse literature on the relation between downside risk and asset prices. In chapter one, we use a amous quote among proessional investors, "ocus on the downside, and the upside will take care o itsel", to motivate a representative consumer-investor who only cares about the downside. The consumption-based asset pricing model that emerges rom this idea explains the main existing puzzles ound within the asset pricing literature. These include the equity premium and the risk-ree rate puzzles, the countercyclicality o the equity premium and the procyclicality o the risk-ree rate. The model is parsimonious, requiring only three preerence-related parameters: the time discount actor, the elasticity o intertemporal substitution, and the downside risk aversion. When we use the model to understand the relation between returns and consumption in the US, we ind that the itted parameter values are consistent with what is expected rom the micro oundations. In chapter two, we show that the model proposed in chapter one can also explain the inancial puzzles in other developed countries. This is an important step in the empirical validation o the model. The estimated parameters are robust across highly capitalized countries and qualitatively close to the ones obtained or the US.

4 Moreover, the risk measure under the quantile utility model can better justiy the dierences in risk premia across countries when compared to the risk measure under the expected utility model. In chapter three, we evaluate the eect o margin requirements on asset prices, an additional channel or the relation between downside risk and prices. We provide evidences o the existence o an aggregate margin-related premium in the economy. In particular, we show that (i) a margin-related actor is able to predict uture excess returns o the S&P 5 and (ii) stocks with high betas on the margin-related actor pay on average higher returns compared those with low margin betas. These result are important not only to understand asset prices, but also the unconventional polices implemented by the Fed during the great recession o Although data on margin requirements or the S&P 5 utures are publicly available, it is in general very hard to obtain inormation on margins or other assets. Given that, we also propose a nonparametric model or estimating margins as a unction o the asset s value at risk. This is theoretically justiiable and has good empirical results.

5 Contents Contents i List o Tables iv List o Figures vi Acknowledgments viii Preace ix Asset Pricing under Quantile Utility Maximization Introduction Quantile utility maximization and asset pricing Elements Asset pricing Dynamics, model solution and simulation Dynamics : constant economic uncertainty Dynamics 2: stochastic economic uncertainty What is a reasonable value or τ? Comparing results i

6 Contents.4 Model estimation A general estimation method A simple two-step estimation procedure Empirical results Conclusion International Data and the Quantile Utility Asset Pricing Model Introduction International data Fitting the model to each country Model estimation Updated data set Conclusion An Empirical Evaluation o the Eects o Margins on Asset Prices Introduction The theoretical model Testable implications o the model Predicting the market portolio return The margin actor and the cross-section o expected returns Further empirical investigation Omitting consumption Modelling margins Results with augmented sample Conclusion ii

7 Contents Bibliography Appendix iii

8 List o Tables. Coniguration o the model parameters or simulation Simulated moments Estimates under no serial-correlation Estimates under serial-correlation International descriptive statistics International parameters under lognormality International risk aversion under expected utility International estimated parameters Updated international descriptive statistics International estimated parameters - updated sample month ahead regressions month ahead regressions month ahead regressions month ahead regressions Controlling or possible isolated eects o the ted spread and margins Cross-sectional regressions with and 25 portolios The eect o ommiting consumption iv

9 List o Tables 3.8 Using itted margins instead o true margins Omitting consumption and using itted margins Descriptive statistics Turbulent period: 2-month ahead regression v

10 List o Figures. The quantile utility agent s reasoning A simple lottery and certainty equivalents Relation between τ and γ Mean excess return X risk under expected utility Mean excess return X risk under expected utility (high capitalized countries) Mean excess return X risk under quantile utility Mean excess return X risk under quantile utility (high capitalized countries) International risk-return trade-o (expected utility) - updated sample International risk-return trade-o (quantile utility) - updated sample The time-series or x t The time-series or m t and ψ t The time-series or x t m t ψ t (level not identiied) The margin-related premium portolios sorted by margin beta Margin x itted margin (in sample) Comparing in and out-o-sample itted values vi

11 List o Figures 3.7 The actor or the augmented sample ( m t ψ t ) t-statistic vs. ted standard deviation (-month ahead regressions) t-statistic vs. actor (m t ψ t ) standard deviation (-month ahead regressions) vii

12 Acknowledgments I would like to thank my advisor Dennis Kristensen or invaluable support on this dissertation. This work would not have been possible without his guidance and suggestions along the way. I also thank Andrew Ang, Pierre-André Chiappori and Marcelo Moreira or constant and crucial advice. I am deeply indebted to them as well. Many other people have directly contributed to this work. Among them, I would like to highlight John Donaldson, Guilherme Martins, Marcos Nakaguma, Ricardo Reis, and Bernard Salanié. I have to express my deepest gratitude to my parents Marcio and Salete, who are the oundation o all my accomplishments. Thank you or everything. This dissertation is dedicated to my wie Priscilla. Thank you so much or all your love, support, company, and encouragement during this time. You are the responsible or this. viii

13 Preace This dissertation contributes to the recent and diverse literature on the relation between downside risk and asset prices. Distinct theoretical models have been developed to understand and ormalize such a relation. For example, Bekaert, Hodrick and Marshall (997), Epstein and Zin (2), Barberis, Huang and Santos (2), Ang, Bekart and Liu (25) and Routledge and Zin (2) address some standard inancial and macroeconomic puzzling acts employing asymmetric preerences (agents somehow overweight bad outcomes when evaluating a risky situation). In a dierent ramework, considering heterogeneous-risk-aversion agents acing margin constraints, Garleanu and Pedersen (29) demonstrate that margin requirements, which are directly related to downside risk, may be relevant to determine prices (Brunnermeier and Pedersen 28, Gromb and Vayanos 2 and Geanakoplos 2 present similar results). In addition, the disaster models constitute another example o the connection between asset prices and downside risk. Barro (26) and Kelly (29), in the same spirit as Reitz (988), show that the mere potential or inrequent extremely bad events can have important eects on asset prices. The dissertation consists o essentially two parts, both o which are related to the literature above. In the irst part, developed in chapters one and two, we contribute to the research on asset pricing under asymmetric preerences. In the second part, developed As Donaldson and Mehra (28) point out, asymmetric preerences models and disasters models can be seen as dual to one another: either agents in the model must eectively be very sensitive to bad outcomes, or it is the outcomes themselves that must be very bad. ix

14 Preace in chapter three and co-authored with Guilherme B. Martins, we investigate the eect o margins on asset prices. In chapter one, we use a amous quote among proessional investors, "ocus on the downside, and the upside will take care o itsel", to motivate a representative consumerinvestor who only cares about the downside. The consumption-based asset pricing model that emerges rom this idea explains the main existing puzzles ound within the asset pricing literature. These include the equity premium and the risk-ree rate puzzles, the countercyclicality o the equity premium and the procyclicality o the risk-ree rate. The model is quite parsimonious, requiring only three preerence-related parameters: the time discount actor, the elasticity o intertemporal substitution, and the downside risk aversion. When we use the model to understand the relation between returns and consumption in the US, we ind that the itted parameter values are consistent with what is expected rom the micro oundations. The parsimony o the model is a relevant characteristic. The good empirical results rom Barberis, Huang and Santos (2) and Routledge and Zin (2) indicate that the consideration o asymmetric preerences over good and bad outcomes is a promising path or theories on choices and, in particular, or a well-accepted resolution o the asset pricing puzzles. Nevertheless, the large number o preerence-related parameters in these models (six and ive, respectively), which is crucial or their success, is a delicate issue. First, it is not easy to translate the models into a comprehensive view o the whole process. Second, it is hard to assign precisely the corresponding importance o each parameter to the obtained results. Finally, and perhaps most problematic, matching data by augmenting the x

15 Preace parametric dimension is subject to the standard over-itting critique. Given its parsimony, the model developed in chapter one addresses all these issues. Chapter two takes chapter one s model to international data. As Campbell (999, 23) shows, the standard inancial puzzles are also present in other developed countries. Hence, we have an opportunity to submit our model to an additional test. Would it be successul i conronted with an international data set? Chapter two presents evidences o a positive answer to this question. By estimating the model or ten developed countries, we obtain reasonable estimates or the risk and intertemporal preerences in general. The estimated parameters are robust across highly capitalized countries and qualitatively close to the ones obtained or the US. We compare our results to Campbell s (23), who estimates the canonical expected utility model or the same countries. Moreover, we show that the risk measure under the quantile utility model can better justiy the dierences in risk premia across countries when compared to the risk measure under the expected utility model. The second part o the dissertation is developed in chapter three. As mentioned above, a number o recent theoretical papers have been suggesting that margins can aect asset prices in periods where risk tolerant agents are credit constrained. The relation between margin requirements and downside risk is straightorward. When an investor buys stocks on margin, some money is put up by him (initial margin), and the remainder is borrowed rom the broker, with the purchased shares used as collateral. How does the broker deine the maximum lending amount? According to the collateral evaluxi

16 Preace ated at a worst-case scenario. Thereore, the worse the worst-case scenario, the higher the initial margin requirement. The result o margins aecting prices would be important not only to understand asset prices per se, but also the unconventional credit policy implemented by the Fed during the great recession o The size and composition o the Fed s balance sheet has suered major changes in the past three years. In January 27, the Fed carried no risk o deault in its assets, holding basically US Treasury bills ($ 78 billion). During the crisis, however, a variety o asset were included in the balance sheet in signiicant amounts. For example, commercial papers ($ 35 billion), repurchase agreements ($ 5 billion), mortgage-backed securities ($ trillion), Federal agency debt securities ($ 5 billion) and others ($ billion). In December 2, the total size o the balance sheet was almost $ 2.5 trillion. As Geanakoplos (2) points out, the negative eect o margins on prices, together with the act that these elements eed back one each other, could justiy such a radical change in the credit policy. According to him, during some periods, "the Fed must step around the banks and lend directly to investors, at more generous collateral levels than the private markets are willing to provide." In addition, the margin premium may break the usual non arbitrage link between the Fed und rate and the rate o returns o other assets, aecting the ability o the monetary authority to promote an expansionary policy. As we shall see in chapter three, the margin premium is the product o the margin requirement, the cost o margin, and the importance o the leveraged agents in aggregate consumption. The cost o margin is equal to the shadow xii

17 Preace cost o capital, which can be measured by the dierence between the uncollateralized and the collateralized short term rates. The latter is closely related to the Fed und rate, while the ormer depends on the liquidity and credit condition in the interbank market. Hence, during a inancial crisis, when margin constraints are binding, a reduction in the Fed und rate may not translate into a all on the rate o returns o other assets. The reason is that the consequently higher shadow cost o capital steepens the margin-return relation, and this increases the required return on assets with high margin requirements. Since in bad periods margins are signiicantly higher across assets, the interest rate reduction can then have small, zero, or even a positive eect on the required return o other assets in the economy. Despite the importance o this result, empirical evidence is still scarce. Chapter three contributes to ill this gap, inding empirical support or the existence o an aggregate margin-related premium. Our empirical indings are related to both the time-series and cross-section o returns. In particular, we show that (i) a margin-related actor is able to predict the uture excess returns o the usual proxy or the market portolio (S&P 5), and (ii) portolios with high betas on the margin actor pay on average higher returns in relation to those with low margin betas. Although data on margin requirements or the S&P 5 utures are publicly available, it is in general very hard to obtain inormation on margins or other assets. Given that, chapter three also proposes a nonparametric model or estimating margins as a unction o the asset s value at risk. This is theoretically justiiable and has good empirical results. xiii

18 Chapter Asset Pricing under Quantile Utility Maximization. Introduction A amous quote among proessional investors is "Focus on the downside, and the upside will take care o itsel". 2 In this paper, we consider a representative consumer-investor who ollows this advice. Surprisingly, the consumption-based asset pricing model that emerges rom this idea explains the main existing puzzles ound within the asset pricing literature. These include the equity premium and the risk-ree rate puzzles, the countercyclicality o the equity premium and the procyclicality o the risk-ree rate. In the proposed model, the consumer-investor is concerned with the so-called downside risk. This is done by replacing the standard setting o expected utility optimizing agents with the concept o quantile utility. Under this ramework, the agent summarizes a risky situation using a worst-case scenario which is a unction o his downside risk aversion. The more downside risk averse the agent, the worse the worst-case scenario he considers. The τ quantile o a continuous random variable can be interpreted as the worst possible outcome that can occur with probability τ. Hence, instead o maximizing the expected value o his utility unction, the agent maximizes a given τ quantile o it. As we will see, τ deines his downside risk aversion: the lower τ, the higher the downside risk aversion. 3 2 A search o this sentence on the internet returns many results. 3 One could say that the agent s objective unction is given by the value at risk (VaR) o his utility. However,

19 . Introduction 2 This is a novel extension o the static decision-theoretical ramework developed by Manski (988) and Rostek (2) or a dynamic asset pricing setting. In a standard economy with one risky and one risk-ree asset, we can derive an arbitrage-ree asset pricing model, where both main characteristics o the canonical expected utility consumptionbased approach (Hansen and Singleton (982), Mehra and Prescott (985), hereinater, the canonical model) are modiied. The equity premium is no longer based on the covariance between the risky return and the consumption growth. Instead, it is a linear unction o the risky return standard deviation. In addition, risk aversion and elasticity o intertemporal substitution (EIS), which are linked throughout a single parameter in the canonical model, are automatically disentangled in a simple way. These two endogenous changes are the main drivers o the good empirical results. Since stock returns historically have a high standard deviation, the price o such a risk, i.e., the level o downside risk aversion, will not have to be high to match the empirical excess returns. Moreover, the attitude towards intertemporal substitution is not polluted by risk preerences. To reproduce (i) the irst and second moments o the risk-ree return, the equity premium, and the consumption growth, (ii) the low covariance between risky return and consumption growth, (iii) the countercyclical risk premium, and (iv) the procyclical risk-ree rate that we see in data, our model requires only three parameters related to preerences: a downside risk aversion (τ) o about.43, an EIS (ψ) o about.5 and a time discount actor (β) o less than. A downside risk aversion o such a magnitude is reasonable in that it since τ here is a ree parameter deining preerence towards risk, it is not restricted to being close to zero (as in standard VaR applications).

20 . Introduction 3 produces reasonable certainty equivalents or bets on continuously distributed random variables (stock indexes, or example). By comparing certainty equivalents under quantile and expected utility maximization, an agent with this level o downside risk aversion is analogous to an expected utility agent with a relative risk aversion coeicient o 3. According to Mehra and Prescott (985) reasonable values or such a parameter would be between and. An EIS o about.5 is also an acceptable value. In a recent work using microdata, Engelhardt and Humar (29) estimate the EIS to be.74, with a 95% conidence interval that ranges rom.37 to.2. Using macrodata and separating stockholders rom nonstockholders, Vissing-Jorgensen (22) estimates the EIS around.4 and.9 or these respective groups. To illustrate the main dierences between the predictions o our ramework and the predictions o the canonical model, we irst derive equations in closed-orm or the risky return, the risk-ree rate, and the equity premium. These equations come rom combining the Euler equations o the quantile agent with the standard assumption o joint lognormality o returns and consumption growth. In order to replicate the well-evidenced existence o predictability in uture excess returns, we then allow or time-varying economic uncertainty in the aggregate economy dynamics. From this, a countercyclical risk premium and a procyclical risk-ree rate are produced. Taking the model to data, we irst perorm simulation exercises, matching the irst and second moments o consumption growth, risk-ree rate and excess returns. Then, to evaluate the model ree o distributional assumptions, we propose a GMM-based estimation method or its parameters.

21 . Introduction 4 The derived Euler equations impose restrictions on the unctional orms o the conditional τ quantiles o consumption growth and excess return. They are well-deined unctions o the period-by-period risk-ree rate and o the other parameters related to preerences. However, as τ in this ramework is not given (it is the downside risk aversion to be estimated), the standard asymptotic results or quantile regressions as a GMM problem do not apply. Hence, we derive suicient conditions or the parameters to be globally identiied and or the proposed estimator to be consistent. The act that the model separates risk and time preerences allows us to estimate the EIS. This is a useul result o this paper. Under the standard technology or disentangling EIS and risk aversion (Epstein and Zin s (989) preerences), one has to use instrumental variables to estimate the EIS. This is what Hall (988) and Campbell (23) do or example. Such estimations were recently ound to suer rom weak-instruments related issues 4 and thereore are not reliable (see Neely, Roy, and Whiteman (2) and Yogo (24), or instance). However, the EIS estimation under our model does not require the use o any instrument. We conclude the introduction by positioning this study in the related literature. The research in asset pricing can be separated according to the modiications proposed with respect to the canonical model. Such modiications are about (i) preerences, (ii) market and asset structure, and (iii) the endowment process. Group (i) could be urther divided into two branches: (i.i) preerences inside and (i.ii) preerences outside the expected utility 4 To estimate the EIS under Epstein and Zin s preerences one has to use instruments or consumption growth or returns. Since both o these variables are only weakly predictable, the instruments are weak.

22 . Introduction 5 ramework. Kocherlakota (996), Cochrane (997, 26), Campbell (999, 23), and Donaldson and Mehra (28a, 28b) provide good surveys o this literature. The current study belongs to branch (i.ii), which was initiated by Epstein and Zin (989) and Weil (989). These authors use the recursive preerences o Kreps and Porteus (978) as a way o separating time and risk preerences, something that is not possible under the canonical model. By disentangling risk aversion and EIS, they end up with a three-parameter model which is able to generate a reasonable level or the risk-ree rate. However, since no innovation in the risk dimension is made, a high level o risk aversion is still necessary to it the equity premium. Epstein and Zin (99, 2) and Bekaert, Hodrick and Marshall (997) investigate the use o Gul s (99) disappointment aversion preerences to explain the equity premium puzzle. 5 According to these preerences, outcomes below the certainty equivalent are overweighted relative to outcomes above it. Although such preerences are a one-parameter extension o the expected utility ramework, these papers extend the canonical model in two parameters, since they also use the model o Epstein and Zin (989) to disentangle risk aversion and EIS. However, they are able to it the equity premium with only a slightly lower, still unreasonable, risk aversion level. 6 Going urther, Routledge and Zin (2) extend the disappointment aversion model in one additional dimension. They generalize Gul s preerences by deining an outcome 5 Single-period portolio allocation is studied under disappointment aversion by Ang, Bekart and Liu (25). Basset, Koenker and Kordas (24) also study sinlge-period allocation using preerences that accentuate the likelihood o the least avorable outcomes. 6 Bonomo and Garcia (993) show that it is crucial to combine Gul s preerences with a joint process or consumption and dividends that ollows a Markov switching model in order to match the irst and second moments o risk-ree and excess returns under reasonable parameter values. However, a model such as that would be in both groups (i.ii) and (iii) deined above.

23 . Introduction 6 as disappointing only when it is suiciently ar (deined by the new parameter) rom the certainty equivalent. Since their model also separates risk aversion and EIS using Epstein and Zin (989) preerences, they are a three-parameter extension o the expected utility model, resulting in a total o ive preerence-related parameters. Under this richer structure, the disappointment aversion-based ramework is inally able to address the inancial puzzles successully. An alternative way o considering the act that people care asymmetrically about good and bad outcomes is provided by the prospect theory o Kahneman and Tversky (979). Applying prospect theory to asset pricing, Barberis, Huang and Santos (2) are also able to reproduce the inancial data patterns under reasonable parameter values. 7 In their model, the representative agent derives direct utility not only rom consumption, but also rom changes in the value o his inancial wealth. Moreover, he is more sensitive to negative movements in his inancial wealth than to positive movements. Besides that, such a sensitivity also is a unction o the agent s past portolio experience: i he had losses in the past relative to a time-varying benchmark, he now is more sensitive to urther losses. A unctional orm relecting this mechanism is imposed by the researchers. Barberis, Huang and Santos s (2) model also employs a large number o preerencerelated parameters; six, to be exact. The irst two are the time discount actor and the relative risk aversion related to consumption. The third is the agent s extra sensitivity to losses in his portolio wealth. The orth deines how previous losses impact the third parameter. The ith determines how the benchmark used by the agent to deine gains and losses 7 Benartzi and Thaler (995) investigate single-period portolio allocation under prospect theory.

24 . Introduction 7 evolves over time. The sixth controls the overall importance o utility rom gains and losses in inancial wealth relative to utility rom consumption. The good empirical results rom Barberis, Huang and Santos (2) and Routledge and Zin (2) indicate that consideration o asymmetric preerences over good and bad outcomes is a promising path or theories on choices and, in particular, or a well-accepted resolution o the asset pricing puzzles. Nevertheless, the large number o preerence-related parameters in these models, which is crucial or their success, is a delicate issue. First, it is not easy to translate the models into a comprehensive view o the whole process. Second, it is hard to assign precisely the corresponding importance o each parameter to the obtained results. Finally, and perhaps most problematic, matching data by augmenting the parametric dimension is subject to the standard over-itting critique. According to this critique, the larger number o parameters may simply describe better the noise in the data, rather than the underlying economic relationships. In other words, these models could be providing spurious data-itting. 8 The present paper addresses these issues. The quantile utility criterion comes rom a loss-unction that asymmetrically weighs good and bad outcomes, the well-known check loss-unction. Hence, the derived model under this ramework belongs to the class o models related to asymmetric preerences. Moreover, the model is quite parsimonious, requiring only three preerence-related parameters: the time discount actor; the EIS; and the 8 This tense relationship between the augmentation o the expected utility ramework with additional parameters and the over-itting critique is raised, or instance, by Zin (22). Based on that article, Watcher (22) claims that "behavioral models leave room or multiple degrees o reedom in the utility unction. Taken to an extreme, this approach could reduce structural modeling to a tautological, data-itting exercise" and "I believe that parsimony lies at the root o what Zin reers to as reasonableness. A parsimonious model is a model in which the number o phenomena to be explained is much greater than the number o ree parameters."

25 .2 Quantile utility maximization and asset pricing 8 downside risk aversion. Finally, it solves the main asset pricing puzzles addressed by Barberis, Huang and Santos (2) and Routledge and Zin (2). The rest o this work is organized as ollows. Section.2 presents the quantile utility agent in its general orm and derives some basic results o asset pricing under quantile maximization. Section.3 solves the model under lognormality and simulates rom it. Section.4 discusses how to estimate the model ree o distributional assumptions and presents the results. Section.5 concludes..2 Quantile utility maximization and asset pricing In this section, we irst present the elements o the quantile utility model, ollowing Manski (988) and Rostek (2). Then, we apply this theoretical-decision ramework to asset pricing..2. Elements A general choice theory or quantile maximizing agents was developed recently. Rostek (2) is the irst study to axiomatize the quantile utility agent. Notwithstanding, the quantile maximization model or decision making under uncertainty was irst proposed 22 years ago by Manski (988). The main idea is simple. An agent, when acing a situation where he has to choose among uncertain alternatives, picks the one that maximizes some given quantile o the utility distribution instead o its mean, as in the expected utility model. In this ramework,

26 .2 Quantile utility maximization and asset pricing 9 the agent cares about the worst outcome that can happen with a given probability. For instance, the given quantile can be the median o the utility distribution, or the.25 quantile. In the case o the.25 quantile or example, when evaluating an uncertain situation, he looks at the worst outcome that can occur with 75 percent probability (i.e., the chance o the realized scenario being better than the scenario he considers is 75 percent). The quantile o concern is an intuitive measure o pessimism. I agent A looks at the worst that may happen in 9 percent o the situations, i.e., quantile., and agent B looks at the worst that may happen in 6 percent o the situations, i.e., quantile.4, we would naturally classiy agent B as more optimistic than agent A : agent A picks a more conservative scenario to summarize the lottery. Figure. illustrates this or a lottery that ollows a normal distribution. Fig... The quantile utility agent s reasoning.

27 .2 Quantile utility maximization and asset pricing As we shall see below, the quantile o concern deines also the agent s downside risk preerence. Hence, downside risk preerence is closely related to our standard notion o optimism-pessimism. (i) Asymmetric preerence Because o the characteristics o his loss-unction, we can say that the quantile agent cares asymmetrically about good and bad outcomes. This intuition comes rom Manski (988), based on the work o Wald (937). Assume that an agent has to evaluate an uncertain situation where U is his utility level which can have dierent values in dierent states o the world. This uncertain situation is represented by the cumulative distribution unction o U; denoted by FU : According to the standard ramework in decision theory introduced by Wald (937), this agent should summarize (evaluate) FU using the criterion! that minimizes the expected value o his loss-unction, i.e., his risk-unction. A possible loss-unction could be the square loss. In this case, he would summarize FU using! Z = arg min (z R Z!2R = z dfu (z) :!)2 dfu (z) R Hence, he would use the expected utility criterion o von Neumann and Morgenstern (944) and Savage (954): This allows us to interpret the expected utility agent as someone who is evenly worried with underpredictions and overpredictions o his utility level in a risky

28 .2 Quantile utility maximization and asset pricing situation and uses squares (L2 norm) to compute the distances between the utility level predictions and realizations. What i the decision maker was asymmetrically worried about under and overpredictions o his uture utility level? We could describe a situation like that by the check loss-unction o Koenker and Basset (978). In this case, he would evaluate FU using! = arg min!2r Z ( R ) jz!j [z <!] + jz!j [z!] dfu (z) = Q (U ) ; where Q (U ) is the th quantile o the random variable U (i FU is continuous, Q (U ) = FU ( )): Thereore, a quantile maximizer can be described as someone who asymmetrically weighs underpredictions and overpredictions o his uture utility level, in the ratio ( )= ; and uses absolute values (L norm) to compute the distances.9 In this case, the agent s evaluation criterion is the th quantile may happen with probability ( o his utility, that is, the worst possible utility level that ): This is the optimal criterion to summarize FU given his asymmetric concern with the upper tails o utility distributions relative to their lower tails. Such an agent could also compute distances under the L2 norm. In this case, his criterion to evaluate FU would be the expectiles o Newey and Powell (987) 9! ( ) = E (U ) + 2 E [(U! ( )) [U <! ( )]] :

29 .2 Quantile utility maximization and asset pricing 2 (ii) Quantile agent deinition We now deine the quantile agent in a more ormal way. Let S be a set o states o the world s 2 S; and X be an arbitrary set o payos x; y 2 X : Then, the agent has to choose among simple acts h : S! X ; which map rom states to payos. Let A be the set o all such acts, and E = 2S be the set o all events. Deine to be a probability measure on E; and u a utility unction over payos u : X! R: For each act, induces a probability distribution over payos, reerred to as a lottery. Given that, let G; H denote the random variables (payos) induced by the acts g; h 2 A, respectively. Finally, deine FG and FH as the lotteries induced by the acts g and h, i.e., the cumulative distribution unctions o G and H, respectively. A decision maker is deined as a -quantile maximizer i there exists a unique [; ] ; a probability measure g 2 on E; and a utility unction u, such that or all g; h 2 A; h, Q (u (G)) > Q (u (H)) : As always, we can think in terms o the lotteries: FG FH, Q (u (G)) Q (u (H)) : (iii) Downside risk aversion For the standard expected utility agent, we may understand risk preerences using the ollowing logic. First we deine riskiness. We say that the lottery FH is riskier than the lottery FG i FG second-order stochastic dominates (SSD) FH (see Rothschield and Stiglitz (97)). FG SSD FH i and only i

30 Then, we deine.2 Quantile utility maximization and asset pricing 3 to be the class o all pairs o lotteries that SSD one another, i.e., = (FG ; FH ) : FG SSD FH g: It is natural to classiy agent A as more risk averse than agent B i or all pairs o distributions in ; whenever B preers a distribution which SSD the other, so does A: Finally, we show that this will be the case i and only i the utility unction o agent A is "more concave" than the utility unction o agent B, i.e., ua (x) = where (ub (x)) ; ( ) is an increasing concave unction. Given that, we conclude that risk-aversion is described by the concavity o the utility unction. Manski (988) and Rostek (2) ollow the same logic to attach the quantile maximizer s attitude toward risk to the quantile he maximizes. The central point is that riskiness is characterized in a dierent way, the so-called downside risk: FH involves more downside risk than FG i FG crosses FH rom below. We say that lottery FG crosses lottery FH rom below i there exists x; y 2 X ; such that FG (y) FG (y) FH (y) or all y < x and FH (y) or all y > x. That is, downside risk is related to the probability o bad outcomes. Just as above, considering the class o all pairs o lotteries with the single-crossing property, = (FG ; FH ) : FG crosses FH rom belowg; we say that individual A is more downside risk averse than individual B i, or all pairs o distributions in ; whenever B preers a distribution which crosses the other rom below, so does A: Given that, we can show that agent A is more downside risk averse than agent B i and only i Z A < B; and x [FH (t) FG (t)] dt, or any x 2 X : I FG and FH have the same mean, and FH has more downside risk than FG ; then FH has also more (second-order stochastic dominance) risk than FG : However, under dierent means, this is not true.

31 .2 Quantile utility maximization and asset pricing then 4 can be deined as the downside risk aversion parameter in the decision model: the lower ; the more downside risk averse the agent. But what role does the concavity o the utility unction play under this ramework? Because o the property o equivariance o quantiles to monotonic transormations, the answer to this question is "none", at least or static decision problems. (iv) Equivariance o quantiles to monotonic transormations A key aspect o the quantile utility model is that static decisions are invariant to any strictly increasing transormation o the utility unction. This is described in Proposition in Manski (988). I m : R! R is a strictly increasing unction, and X is a random variable; then2 Q (m (X)) = m (Q (X)) : (.) Hence, or lotteries FG and FH ; FG FH, Q (u (G)), u Q (u (H)) (Q (u (G))), Q (G) u (Q (u (H))) Q (H) ; where the second line ollows rom the act that u is a strictly increasing unction. 2 The intution under this result is that a strictly increasing transormation o the random variables doesn t change the order o the values o their support.

32 .2 Quantile utility maximization and asset pricing 5 Thereore, or static problems, the agent s decision does not depend on u. Manski (988) and Rostek (2) reer to this as a robustness property: the choice is unaected by misspeciication o the utility unction. However, the utility unction is relevant in intertemporal choices. When the utility unction has more than one argument, it is not possible to use the equivariance property to get rid o u. In particular, under time-separability, the concavity o the utility unction deines the preerence towards intertemporal substitution as usual. This is going to play an important role in the asset pricing theory, allowing the downside risk aversion and the EIS to be disentangled. This idea is not in Manski (988) or in Rostek (2) and, to the best o our knowledge, is explored or the irst time in the present study..2.2 Asset pricing We now apply the quantile maximization decision theory to the standard intertemporal problem o a consumer-investor agent. First, we deine the consumption-investment problem and solve or the Euler equations that the agent must respect in equilibrium. Then we discuss the Law o One Price and the no-arbitrage condition under this ramework. The model to be considered has 2 periods. As Karni and Schmeidler (99) show, once we depart rom expected utility, one o the ollowing three assumptions has to be relaxed: (i) time consistency; (ii) consequentialism; or, (iii) reduction o compounded lotteries. Assumptions (i) and (ii) are in the heart o the Principle o Optimality o dynamic programming (see Rust (26), section 3.6). Thereore, to be able to solve a multipleperiod problem outside o the expected utility ramework by standard dynamic program-

33 .2 Quantile utility maximization and asset pricing 6 ming, one must relax assumption (iii). However, by relaxing (iii), one would be including preerences about the time o resolution o the uncertainty in the model, just as in the recursive preerences o Kreps and Porteus (978) and Epstein and Zin (989).3 Since the central goal o this study is to develop a simple, parsimonious and stylized model to address the over-itting critique within the asymmetric preerences literature, we restrict the model to a 2-period ramework. The economy has two assets, one risky and one risk-ree. Deine the value o the risky asset at t + to be Xt+ = Pt+ + Dt+ ; where Pt+ is the price o the asset at t + and Dt+ is the value o some cash low the investor received between t and t + (in the case o a stock, D is the dividend). Deine Xt+ to be the value o the risk-ree asset at t + and Pt its price at t: Let Ct be the agent s consumption at t; and be the quantity o the risky and risk-ree assets he buys at t respectively, and Wt be his initial wealth. Then, under time-separability, he solves: M ax Qt (u (Ct ) + u (Ct+ )) ; s:t: Ct = Wt Pt Pt Ct+ = Xt+ + Xt+ where (.2) 2R2 is the time discount actor, u is the utility unction, Qt (x) is the th quantile o the conditional distribution o the random variable x (conditional on the inormation set available at time t): 3 Indeed, according to Rust (26), recursive preerence is the only class o non-expected utility preerences that allows the use o standard dynammic programming (backward induction) to solve multi-period problems.

34 .2 Quantile utility maximization and asset pricing 7 This agent derives utility only rom consumption, as usual, and cares about the worst outcome (in terms o the utility or both periods) that may occur with probability ( ): In other words, this agent ollows the amous advice "Focus on the downside, and the upside will take care o itsel". As discussed in sub-section.2., the higher his level o downside risk aversion, the lower. A key eature o problem (.2) is that downside risk aversion and elasticity o intertemporal substitution (EIS) are automatically disentangled. This is a direct consequence o the quantile s equivariance or monotonic transormations. Note that, according to equation (.), we have Qt (u (Ct ) + u (Ct+ )) = u (Ct ) + u (Qt (Ct+ )) ; since u is a strictly increasing unction. Hence, all uncertainty in problem (.2) is resolved by parameter, since Qt (Ct+ ) is deterministic at t: The only role played by u is to discount consumption across time: depending on the concavity o u; the agent will combine present consumption, Ct ; and the certainty equivalent o uture consumption (which, or the quantile maximizer, is equal to Qt (Ct+ )): Hence, the concavity o u will only deine the EIS, denoted by : Specializing u (c) = c ; we have =.4 Note that such an assumption or the unctional orm o u imposes no restriction on risk preerence: it simply restricts the EIS to being constant. 4 Deining U (Ct ; Qt (Ct+ )) = we have that Ct + (Qt (Ct+ ))

35 .2 Quantile utility maximization and asset pricing The EIS parameter, = 8 ; deines the degree o substitutability-complementarity between consumption today, Ct ; and the certainty equivalent o consumption tomorrow, Qt (Ct+ ) : For! ; Ct and Qt (Ct+ ) become perect complements, and we have the agent s object unction given by U (Ct ; Qt (Ct+ )) = min Ct ; Qt (Ct+ )g : At the other extreme, or! ; Ct and Qt (Ct+ ) become perect substitutes, i.e., the agent maximizes U (Ct ; Qt (Ct+ )) = Ct + Qt (Ct+ ) : For the intermediate case o = ; we end up with the Cobb-Douglas U (Ct ; Qt (Ct+ )) = Ct (Qt (Ct+ )) : With respect to the time discount actor ; its role is to determine the marginal rate o substitution between Ct and Qt (Ct+ ). Thereore, complementarity between Ct and Qt (Ct+ ) ; and deines the degree o substitutabilityparameterizes such a relation.5 What are the implications o the quantile maximization asset pricing model? With the ollowing proposition, proved in the appendix, we initiate @Qt (Ct+ ) d (Qt (Ct+ ) =Ct ) Qt (Ct+ ) =Ct @Qt (Ct+ ) = : On the empirical side, we will see that both parameters are also separately identiied by our estimation method.

36 .2 Quantile utility maximization and asset pricing Proposition Suppose a consumer-investor solves problem (.2) and u (c) = 9 c. Then, the Euler equations are given by Pt = Ct+ Ct Qt Pt = Qt Qt (Xt+ ) (.3) Xt+ (.4) Ct+ Ct From now to the end o section.3, we study the asset pricing implications o equations (.3) and (.4). The irst step is to understand whether they respect the Law o One Price and the no-arbitrage condition. Then, we solve the model under the standard assumption o joint lognormality or returns and consumption growth, deriving closed-orms or the risky return, the risk-ree rate and the equity premium in equilibrium. Since we ignore transaction costs, any candidate or an equilibrium pricing system has to respect the Law o One Price: prices should be linear. That is, denoting t; t t = to be a portolio ormed at t; with price given by Pt, the pricing system has to imply Pt = t Pt + t Pt : Otherwise, Pt and Pt cannot be equilibrium prices because o arbitrage opportunities among the individual assets and the portolio. Equations (.3) and (.4) respect this condition. Deining t = Qt Ct+ Ct ; we have

37 .2 Quantile utility maximization and asset pricing Pt t Xt+ + t Xt+ = t Qt = t = t Qt ( t Xt+ ) + t t Xt+ = t Qt ( t Xt+ ) + t Qt = t Pt Qt ( t Xt+ ) t Xt+ t Xt+ t Pt ; where the second line ollows rom the quantile equivariance. Note that or a degenerate random variable x, Q (x) = x or any = Xt+ : 2 [; ], and this implies Qt Xt+ As is well-known, a linear pricing system does not completely rule out arbitrage opportunities. Hence, we need to impose two mild conditions to end up with an arbitrageree model. Proposition 2 Suppose that (i) the risky asset payo Xt+ is a continuous random variable and (ii) 2 (; ) : Then, the pricing model given by equations (.3) and (.4) rules out arbitrage opportunities. Both conditions o proposition 2 (proved in the appendix) are reasonable. The continuity o the risky asset payo comes or ree or stock prices. The second condition, more subtle, rules out two well known agents in decision theory, the so-called MaxMin and MaxMax. The MaxMin agent ( = ) summarizes a lottery by looking at the very worst case scenario that may take place (that is, the worst case scenario that may occur with probability ). On the other hand, the MaxMax ( = ) summarizes a lottery by looking at the very best case scenario that may take place (or, in other words, the worst case scenario that may

38 .3 Dynamics, model solution and simulation 2 occur with probability ). Since both agents represent extreme behaviors (the extremely pessimistic and the extremely optimistic), excluding them is not a restrictive assumption..3 Dynamics, model solution and simulation We now solve the model in closed-orm, under joint lognormality o returns and consumption growth, with both constant and luctuating economic uncertainty. Although the solution under constant economic uncertainty is enough to match both the risk-ree rate and the risk premium under reasonable levels or the preerence-related parameters, it does not generate a time-varying risk premium. To improve the model in this direction, we allow stochastic volatility in the economy dynamics. The model is then simulated under this richer environment..3. Dynamics : constant economic uncertainty Assume g t+ = µ c + η t+, η t+ iid N ( ), σ 2 c r t+ = µ r + u t+, u t+ iid N ( ), σ 2 r (.5) where g t+ = log (C t+ /C t ), r t+ = log (X t+ /P t ) and Cov ( η t+, u t+ ) = σcr. Under this ramework, the closed-orms or the risky return, the risk-ree rate and the equity premium are given by the ollowing proposition.