Information Inertia. June 6, Abstract

Transcription

1 Information Inertia Scott Condie Jayant Ganguli Philipp Karl Illeditsch June 6, 2012 Abstract We study how information about an asset affects optimal portfolios and equilibrium asset prices when investors are not sure about the model that predicts future asset values and thus treat the information as ambiguous. We show that this ambiguity may lead to asset demand that is insensitive to changes in news and investors use only their prior information when making portfolio decisions. This insensitivity to news is more severe when prices deviate a lot from its unconditional mean and in contrast to other ambiguity models it also occurs when demand is sensitive to changes in the price. In equilibrium, we show that stock prices may not react to public information that is worse than expected even though there are no information processing costs or other market frictions. The severity of this mispricing depends on the risk of the stock and the magnitude of the news surprise. Keywords: Ambiguity Aversion, Knightian Uncertainty, Informational Efficiency, Information Inertia, Inattention to News, Public Information. JEL Classification: D80, D81, G10, G11, G12. We would like to thank Kerry Back, James Dow, Christian Heyerdahl-Larsen, Sujoy Mukerji, Emre Ozdenoren, Han Ozsoylev, seminar participants at the Wharton Brown Bag Seminar, Oxford University, London Business School, and participants at the European workshop on general equilibrium theory at the University of Exeter for helpful comments and suggestions. Department of Economics, Brigham Young University, Provo, UT Telephone: , Fax: , ssc@byu.edu School of Economics, University of Nottingham, University Park, Nottingham NG7 2RD, UK. Telephone: +44 (0) , Fax: +44 (0) , jayant.ganguli@nottingham.ac.uk Finance Department, The Wharton School, University of Pennsylvania, 3620 Locust Walk, 2426 SH-DH, Philadelphia, PA , Telephone: , Fax: , pille@wharton.upenn.edu 1

2 There is a vast amount of empirical research which documents that returns for many asset classes around the world are predictable. 1 The economic and statistical significance of the predictability results vary from study to study and the strength of these results as well as the theoretical underpinnings and interpretations are widely debated. In this paper, we study how information about an asset affects optimal portfolios and equilibrium asset prices when investors are not sure about the model that predicts future asset values and thus treat the information as ambiguous. We show that this ambiguity leads investors to ignore some information and hence asset prices in equilibrium do not always reflect all available public information about an asset. We refer to this phenomenon as information inertia. Suppose investors receive a public signal and its correlation with the future value of an asset is unknown. Investors are ambiguous (uncertain in the sense of Knight (1921)) about the predictability of future asset values and consider a range of correlations when processing this signal. Specifically, investors consider the correlation that leads to the lowest expected utility for each portfolio they evaluate. This max-min formulation of preferences is axiomatized in Gilboa and Schmeidler (1989) and is a commonly used representation of decisions-making under ambiguity in asset markets, as discussed in Epstein and Schneider (2010). 2 We show that ambiguity about the predictability of future asset values may cause inattention to news and lead investors to use only their prior information when making portfolio decisions. Specifically, there is a range of prices for which investors ignore bad news when they are long in the asset and they ignore good news when they are short in the asset. This insensitivity of demand to news occurs even for risky portfolios that are sensitive to changes in the price in contrast to previous work on ambiguous information where it only occurs with portfolio inertia (Condie and Ganguli (2011a) and Illeditsch (2011)). Moreover, we show that it is more severe when prices deviate 1 For a review of this literature see Cochrane (2005) or Koijen and Nieuwerburgh (2011) and the references therein. 2 These preferences imply behavior that is consistent with experimental evidence (Ellsberg (1961)) and more recent portfolio choice experiments (Ahn, Choi, Gale, and Kariv (2010) and Bossaerts, Ghirardato, Guarnaschelli, and Zame (2010)). 2

3 a lot from its unconditional mean. Investors inattention to some news implies that stock prices fail to incorporate all publicly available information in equilibrium. While good news is always reflected in the stock price some bad news is not. on information processing costs or other market frictions. 3 This mispricing of news does not rely We also show that the reaction of the stock price to news depends on its risk. Risky stocks are more likely to underreact to bad news whereas stocks that are not very risky tend to overreact to this news. However, the most striking result is that stocks with moderate risk show almost no reaction to bad signals even though there is no ambiguity about the fact that the signal is positively correlated with the asset. This paper complements recent work on optimal portfolios and equilibrium asset prices when investors process public signals. Caskey (2009) shows that prices do not reflect all available signals because ambiguity averse investors may prefer a less informative aggregate signal to its more ambiguous components. In contrast to his paper, we focus on one public signal about the future value of an asset and show that the apparent mispricing of this signal depends on the risk of the asset and the magnitude of the news surprise. 4 Epstein and Schneider (2008) show that investors react more to bad signals than to good signals when there is ambiguity about the precision of these signals. Illeditsch (2011) shows that this ambiguity leads to risky portfolios that are insensitive to changes in the stock price. However, these portfolios are sensitive to changes in the signal and thus in equilibrium prices always reflect all available information. This paper contributes to the literature on optimal portfolio choice with ambiguity. We know from Dow and Werlang (1994), Cao, Wang, and Zhang (2005), Epstein and Schneider (2007), Easley and O Hara (2009), and Campanale (2011) that am- 3 There is a growing literature in Finance and Economics that imposes an exogenous bound or cost on the ability of investors to process information (see Sims (2006), Sims (2010), or Veldkamp (2011) and the references therein). 4 Other papers who study the effects of private signals on the informational efficiency of prices in the presence of ambiguity are Condie and Ganguli (2011a), Condie and Ganguli (2011b), Mele and Sangiorgi (2011), Ozsoylev and Werner (2011), and Tallon (1998). 3

4 biguity leads to portfolio inertia at certainty and thus can help explain nonmarket participation. Epstein and Wang (1994), Epstein and Schneider (2010), and Illeditsch (2011) show that portfolio inertia can also arise for risky portfolios which can help explain why many investors who own stock do not show much trading activity. Garlappi, Uppal, and Wang (2007) characterize optimal portfolios with multiple ambiguous assets. Uppal and Wang (2003), Benigno and Nistico (2012), and Boyle, Garlappi, Uppal, and Wang (2012) show that ambiguity leads to under-diversified portfolios and thus can help explain the international home bias puzzle. We show that if there is ambiguity about the predictability of future asset values, then investors use their prior information when contemplating a long (short) position with moderate risk instead of relying on an ambiguous signal that conveys bad (good) news. Our paper is also related to recent literature on portfolio choice and asset pricing when there is ambiguity about the predictability of future asset returns/cash flows. Hansen and Sargent (2010a) study the price of risk when investors who seek robust decision rules find it difficult to differentiate between i.i.d. consumption growth and one with a persistent component (long run risk of Bansal and Yaron (2004)). 5 Chen, Ju, and Miao (2011) solve a dynamic consumption and portfolio choice problem when there is ambiguity about whether stock returns are IID or predictable. Ju and Miao (2012) and Collard, Mukerji, Sheppard, and Tallon (2011) explain many asset pricing puzzles by introducing ambiguity into a dynamic representative agent model in which consumption and dividends follow a hidden state regime-switching process and a hidden state model with a persistent latent state variable, respectively. The first paper considers the robust control approach and the other three papers consider the recursive smooth ambiguity model to describe preferences. 6 Our focus in this paper is on non-smooth preferences which are a good description for ambiguity averse behavior as shown by Ahn, Choi, Gale, and Kariv (2010) and Bossaerts, Ghirardato, 5 For a survey of learning models when investors seek robust decision rules see Hansen and Sargent (2007). 6 Strzalecki (2011) and Maccheroni, Marinacci, and Rustichini (2006) provide axiomatic foundations for the robust control model and Klibanoff, Marinacci, and Mukerji (2005), Nau (2006), Klibanoff, Marinacci, and Mukerji (2009), and Hayashi and Miao (2011) provide axiomatic foundations for the smooth ambiguity model and its dynamic extension. 4

5 Guarnaschelli, and Zame (2010). 7 The rest of the paper is organized as follows. In Section I, we introduce the model. In Section II, we solve for optimal demand and discuss the information inertia results. In Section, III we solve for the equilibrium stock price and discuss the mispricing results, and in Section IV we show that our results are robust to aggregation. I Model Suppose there are two dates 0 and 1. Investors can invest in a risk-free asset and a risky asset. Let p denote the price of the risky asset, d the future value or dividend of the risky asset, and θ the number of shares invested in the risky asset. There is no consumption at date zero. The risk-free asset is used as numeraire, so the risk-free rate is zero. Hence, future wealth w is given by w = w 0 + ( d p ) θ, (1) in which w 0 denotes initial wealth. Suppose investors receive a signal s about the future value of the asset d. Wefocus in this paper on ambiguity averse investors in the sense of Gilboa and Schmeidler (1989). Let u( ) denote the Bernoulli utility function of the investor and define a model m as a joint distribution of d and s. An ambiguity averse investor in the sense of Gilboa and Schmeidler (1989) chooses a portfolio θ to maximize inf E m [u ( w) s = s], (2) m M where M denotes the set of all models considered by the investor and E m [ ] denotes the expectation with respect to the belief generated by the model m. 7 For a discussion of different preferences specifications that describe aversion to ambiguity see Backus, Routledge, and Zin (2004), Epstein and Schneider (2010), and Hansen and Sargent (2010b). 5

6 Suppose that the joint distribution of d and s is normal: d N d, σ2 d ρσ d σ s. (3) s s ρσ d σ s σs 2 Investors are ambiguous about the correlation between the signal and the dividend and thus ρ [ρ a,ρ b ] with ρ a > 0andρ b < 1. 8 We follow Gilboa and Schmeidler (1993) and assume that investors update their beliefs model by model using Bayes rule. 9 Hence, standard normal-normal updating for each ρ [ρ a,ρ b ]leadsto ( d s = s N ρ d + σ ) d ρ (s s),σd 2 σ (1 ρ2 ). (4) s The utility of an investor who holds θ shares of the risky asset is ( ( ) ) ] min E ρ [u w 0 + d p θ s = s. (5) ρ [ρ a,ρ b ] Investors are more averse to ambiguity if the interval [ρ a,ρ b ] is large and therefore the degree of aversion to ambiguous information can be measured by ρ b ρ a. II Optimal Demand In this section, we determine the optimal portfolio of investors who are ambiguous about the predictability of the future value of an asset. We show that investors with long positions do not always react to news that is worse than expected and investors with short positions do not always react to news that is better than expected. We refer to this insensitivity to news as information inertia. γ w Suppose investors have CARA utility over future wealth w (i.e. u( w) = e with γ>0) and let CE(θ) denote the certainty equivalent of an ambiguity averse 8 If ρ b < 0, then replace the signal s with s and the interval [ρ a,ρ b ]with[ ρ b, ρ a ]. The results for the case when 0 [ρ a,ρ b ] are not reported but are available from the authors upon request. 9 See Epstein and Schneider (2003) and Epstein and Schneider (2007) for learning under ambiguity. 6

7 investor. Then the investor s utility given in equation (5) is equal to u (CE(θ)) with CE(θ) = ( min E ρ [ w s = s] 1 ) ρ [ρ a,ρ b ] 2 γvar ρ [ w s = s]. (6) The assumption of CARA utility and normal beliefs lead to mean-variance preferences over future wealth in which the posterior mean is a linear function and the residual variance is a quadratic function of the correlation ρ. Ambiguity averse investors are worried about the effects of ρ on the mean and variance of future wealth and thus consider for each portfolio θ and signal realization s the minimum expected value of future utility (certainty equivalent). Hence, their worst case scenario belief will depend on the portfolio θ and signal s as the next proposition shows. Proposition 1 (Preferences). Let ˆθ a ( s s)/(γσ d σ s ρ a ) and ˆθ b ( s s)/(γσ d σ s ρ b ). The certainty equivalent of an investor with risk aversion γ and aversion to ambiguity described by [ρ a,ρ b ] is ) E ρa [ w s = s] 1γVar 2 ρ a [ w s = s] if θ min (ˆθa, 0 ( ) 2 ) ) E[ w] 1 1 s s γvar [ w] 2 2γ σ s if min (ˆθa, 0 <θ min (ˆθb, 0 ) ) CE(θ) = E ρb [ w s = s] 1γVar 2 ρ b [ w s = s] if min (ˆθb, 0 <θ max (ˆθb, 0 ( ) 2 ) ) E[ w] 1 1 s s γvar [ w] 2 2γ σ s if max (ˆθb, 0 <θ max (ˆθa, 0 ) E ρa [ w s = s] 1γVar 2 ρ a [ w s = s] if θ>max (ˆθa, 0. (7) The certainty equivalent CE(θ) is a continuous and concave function of the stock demand θ. Moreover, it is continuously differentiable except for the portfolio θ =0if s s. Investors who are contemplating a long position in the asset are worried about bad signals with a high ρ and good signals with a low ρ because informative bad signals significantly lower the posterior asset mean whereas good signals that are not very informative only moderately increase the posterior asset mean. Similarly, investors who are contemplating a short position in the asset fear high posterior asset means and thus are worried about good signals that have a high ρ and bad signals that have a low ρ. On the other hand, investors who are contemplating either a long or short position in the asset always fear risk and thus are worried about signals with a low ρ. 7

8 Investors are more worried about the posterior mean for small risks and are more worried about the residual variance for big risks. Hence, investors treat bad signals as informative for moderate long positions in the asset (0 <θ ˆθ b ) and as not very informative for very large long positions (θ >ˆθ a ). 10 There is a range of portfolio positions (ˆθ b <θ ˆθ a ) for which investors beliefs balance the counteracting mean and variance effects. Specifically, ρ (θ) argmin CE S (θ, ρ) = s s 1 ρ [ρ a,ρ b ] γσ d σ s θ, (8) where CE S (θ, ρ) denotes the certainty equivalent of a standard expected utility maximizer in the sense of Savage (1954) with belief ρ. Investors hedge against ambiguity by immunizing utility against changes in the belief ρ and thus CE S (θ, ρ (θ))/ ρ = 0. This leads to marginal utility that is not affected by changes in the signal. It follows from the Envelope Theorem that CE(θ) θ = CES (θ, ρ (θ)) + CES (θ, ρ (θ)) ρ (θ) θ ρ θ = CES (θ, ρ (θ)) ] [ ] =E[ d p θγvar d. θ (9) We will show below that this desire to hedge against ambiguity leads to demand that is insensitive to changes in news. A Savage Benchmark The demand of a standard expected utility maximizer in the sense of Savage (1954) with belief ρ is given in the next proposition. 11 Proposition 2 (Optimal Demand Savage Benchmark). Let ρ a = ρ b = ρ. Then the 10 If investors contemplate a short position in the asset, then they treat bad signals as not very informativebecausetheworstcasescenariofor the posterior mean and variance is a low ρ. 11 The proof is straightforward and thus omitted. 8

9 demand function for the risky asset is θ ρ (p) = E ρ [ d s = s ] p γvar ρ [ d s = s ]. (10) An increase in the signal will always lead to an increase in the stock position. This is no longer true when investors are ambiguous about the correlation between the signal and the dividend as the next theorem shows. B Ambiguity-Averse Investors Let θ a (p) denote the demand of a Savage investor with belief ρ a, θ 0 (p) denotethe demand of a Savage investor with belief ρ =0,andθ b (p) the demand of a Savage investor with belief ρ b,andθ(p) the demand of an ambiguity averse investor. The optimal demand for an ambiguity averse investor is given in the next proposition. Theorem 1 (Optimal Demand). Let ˆθ a ( s s)/(γσ d σ s ρ a ) and ˆθ b ( s s)/(γσ d σ s ρ b ). Optimal demand for an investor with risk aversion γ and aversion to ambiguity described by [ρ a,ρ b ] is θ(p) = ) θ a (p) p p 1 μ ρa γv ρa max (ˆθa, 0 ) max (θ 0 (p), 0) p 1 < p p 2 μ ρb γv ρb max (ˆθb, 0 ) θ b (p) p 2 < p p 3 μ ρb γv ρb min (ˆθb, 0 ) min (θ 0 (p), 0) p 3 < p p 4 μ ρa γv ρa min (ˆθa, 0 θ a (p) p > p 4, where μ ρ (s) = d + σ d σ s ρ (s s) and v ρ σ 2 d (1 ρ2 ). (11) Suppose the signal conveys bad news (see left graph of Figure 1). 12 Ambiguity averse investors (solid line) are more worried about risk than a low posterior mean when they take on large long positions in the asset and hence their demand coincides with the demand of a Savage investor with belief ρ a (blue dashed line). However, they are more worried about a low posterior mean than risk if they take on a moderate 12 Optimal demand when the signal conveys good news is shown in the right graph of Figure 1. 9

10 long position and hence their demand coincides with the demand of a Savage investor with belief ρ b (chain-dotted line) Bad News ρ a = Good News ρ a = 0.5 ρ b = 0.9 ρ b = ρ = 0 [ ρ a, ρ b ] = [ 0.5, 0.9 ] ρ = 0 [ ρ a, ρ b ] = [ 0.5, 0.9 ] 0 Demand 0.5 Demand Price Price Figure 1: Optimal Demand The left graph shows optimal demand when the signal conveys bad news (s =90< s = 100) and the right graph shows optimal demand when the signal conveys good news (s = 110 > s = 100). Demand is plotted as a function of the stock price for a Savage investor with belief ρ b (red chain dotted line), for a Savage investor with belief ρ a (blue dashed line), for a Savage investor with belief ρ = 0 (green dotted line), and for an ambiguity averse investor with range of beliefs [ρ a,ρ b ] (black solid line). There is a range of prices over which ambiguity averse investors are neither long nor short the stock. Moreover, there is a range of prices for which ambiguity averse investors adjust their stock position but do not react to news. The parameters are d = s = 100, σ d = σ s =5,ρ a =5/10, ρ b =9/10, and γ =1. It is well known that hedging against ambiguity leads to portfolio inertia at the risk-free portfolio because investors who are contemplating a long (short) position in the asset are worried about a low (high) posterior asset mean and thus fear bad (good) signals with a high ρ and good (bad) signals with a low ρ. Hence, there is a range of prices for which they would rather not be invested in the stock (see Dow and Werlang (1992)). What is new in this paper is that there is a range of prices for which investors hedge against ambiguity by choosing a risky portfolio that does not depend on the 13 If ambiguity averse investors take on a short position in the asset, then they behave like a Savage investor with belief ρ a because in this case the worst case scenario for the posterior asset mean and residual asset variances is a low ρ. 10

11 signal. In this case their demand coincides with the demand of a Savage investor who thinks the correlation between the asset and the signal is zero. This is the case even though there is no ambiguity about the fact that the signal is positively correlated with the asset. However, the utility of the ambiguity averse investor is not the same as the utility of a Savage investor with belief ρ = 0 because hedging against ambiguity comes at an utility cost (see Proposition 1). There is a range of long positions that do not depend on signals that convey bad news because investors want to avoid ambiguity. Similarly, there is a range of short position that do not depend on signals that convey good news. Hence, hedging against ambiguity leads to risky portfolios that are insensitive to changes in the signal. 14 These portfolios are more risky for large news surprises. Moreover, the range of prices for which investors hedge against ambiguity by choosing a risky portfolio that does not depend on the signal is increasing in the news surprise as the next proposition shows. 15 Proposition 3. The size of the price region for which investors have downward sloping demand which does not depend on the signal is σ d ( 1 ρ a 1 ρ b ) s s σ s. (12) We now summarize the predictions for optimal portfolios. Model Predictions 1 (Portfolio Choice). If investors are ambiguous about the predictability of future asset values, then (i) there is a range of prices over which investors adjust their long stock position but do not react to news that is worse than expected, (ii) there is a range of prices over which investors adjust their short stock position but do not react to news that are better than expected, 14 There is no portfolio inertia away from certainty because there is no risky portfolio that is independent of the correlation ρ and perfectly hedges against ambiguity by making utility independent of the correlation ρ. Illeditsch (2011) assumes ambiguity about the precision of the signal and shows that there is a risky portfolio that perfectly hedges against ambiguity and thus causes portfolio inertia away from certainty. However, this portfolio is sensitive to changes in the signal and thus there is no information inertia. 15 The size of the price region for which investors are neither long nor short the asset is reported in Proposition 8 of the appendix. 11

12 (iii) there is a range of prices over which investors are neither long nor short the stock when news is surprising, and (iv) the inaction regions are larger for extreme news surprises. C Information Inertia So far we have only looked at how price changes affect the optimal demand for the risky asset. We now take prices as given and ask how changes in the signal affect asset demand. We know that for a Savage investor with belief ρ demand is strictly increasing in the signal. Moreover, there is a strong price reaction to news if ρ is high and/or risk aversion is low. Specifically, for every price p and correlation 0 < ρ < 1 we have that θ(s) s = 1 ρ γσ d σ s 1 ρ. 2 If investors are ambiguous about the correlation ρ and the unconditional risk premium of the stock is non-zero (p d), then there are two ranges of signals for which demand is insensitive to changes in the signal. The left graph of Figure 2 shows optimal demand as a function of the signal when the unconditional risk premium is positive (p < d) and the right graph of Figure 2 shows it when the unconditional risk premium is negative (p > d). 12

13 Positive Unconditional Risk Premium Negative Unconditional Risk Premium 0.6 Dbar p = 10 Dbar p = 5 Dbar p = Dbar p = 10 Dbar p = 5 Dbar p = Demand 0 Demand Signal Signal Figure 2: Information Inertia in Demand The left graph shows optimal demand as a function of the signal when the unconditional risk premium is positive and the right graph shows it when the unconditional risk premium is negative. Both graphs show that when the price deviates a lot from its unconditional mean, then information inertia is more severe and the nonzero portfolios at which investors exhibit information inertia are more risky. The parameters are d = s = 100, σ d = σ s =5,ρ a =5/10, ρ b =9/10, and γ =1. There is no information inertia when the unconditional risk premium is zero (black solid line) because in this case the ambiguity averse investor behaves like a Savage investor with belief ρ a. Intuitively, investors will long the asset when news are good and they will short the asset when news are bad. But there is no confusion about the interpretation of the signal when news are good (bad) and investors are long (short) the asset because the worst case scenario for the posterior mean and the posterior variance is a low ρ. Suppose the unconditional risk premium is positive (blue dashed line and red chain-dotted line in the left graph of Figure 2). If the signal conveys good news, then investors buy the asset and the worst case scenario for both the mean and the variance is always a low ρ. This is no longer true when the signal conveys bad news. Specifically, for a moderate bad news surprise, investors are still long in the asset and thus behave like a Savage investor with belief ρ b if they are more worried about a low posterior mean and they behave like a Savage investor with belief ρ a if they are more worried about risk. On the other hand, if news is very bad then investors take on a 13

14 short position in the asset and thus always behave like a Savage investor with belief ρ a because the worst case scenario for the posterior mean and variance is a low ρ. There are two ranges of signal values for which demand is insensitive to changes in the news. The first range corresponds to the well known case of portfolio inertia at certainty (θ = 0) whereas the second range corresponds to risky portfolios (θ 0) for which investors do not exhibit portfolio inertia. Intuitively, investors exhibit information inertia when they exhibit portfolio inertia at certainty because a small increase (decrease) in the signal does not sufficiently raise (lower) the posterior asset mean to convince an investor to give up a portfolio that perfectly hedges against risk and ambiguity. 16 What is new in this paper is the second range of signal values for which investors change their risky portfolio in response to changes in the stock price but not in response to changes in the signal. This information inertia is more severe when price deviates a lot from its unconditional mean. However, the probability of having information inertia is not monotone in the unconditional risk premium. We determine in the next proposition the size of the signal inaction region and the probability of having information inertia. 17 Proposition 4 (Information Inertia for Risky Portfolios). Thesizeofthesignal region for which investors are either long ( d >p) or short ( d <p) the asset and demand does not react to changes in the signal is σ s σ d (ρ b ρ a ) d p. (13) The probability of investors exhibiting information inertia for either a long ( d >p) or a short ( d <p) position in the asset is ( N p 0,1 ρ d ) ( p a N 0,1 ρ d ) b, (14) σ d σ d where N 0,1 ( ) denotes the cumulative distribution function of a standard normal distributed variable. 16 This form of inertia appears in Condie and Ganguli (2011a) and Illeditsch (2011). 17 The size of the signal region for which investors are neither long nor short the asset and the probability of having information inertia in this case is reported in Proposition 9 of the appendix. 14

15 The probability of having information inertia depends on the unconditional risk premium and is plotted in Figure 3. If the unconditional risk premium is zero, then there is no information inertia because investors behave like Savage investors with belief ρ a. The probability of investors taking on a large long position in the asset increases with the unconditional risk premium and thus the probability of having information inertia decreases because investors with very risky positions are more worried about risk than a low posterior asset mean and thus behave like Savage investors with belief ρ a. The intuition is similar for investors with large short positions. Hence, the probability of short sellers, asset buyers, and investors who do not participate in the stock market to exhibit information inertia is non monotonic in the unconditional risk premium of the asset Positive Unconditional Risk Premium θ < 0 θ = 0 θ > 0 total Negative Unconditional Risk Premium θ < 0 θ = 0 θ > 0 total Probability Probability Unconditional Risk Premium Unconditional Risk Premium Figure 3: Probability of Information Inertia in Demand Both graphs show the probability of having information inertia in demand as a function of the unconditional risk premium. The blue dashed line shows the probability of having inertia for a short position in the stock, the red chain-dotted line shows it for a long position in the stock, the green dotted line shows it for the risk-free portfolio, and the black solid line shows the total probability of having information inertia in demand. The parameters are d = s = 100, σ d = σ s =5,ρ a =5/10, ρ b =9/10, and γ =1. We conclude this section with a summary of the information inertia results. Model Predictions 2 (Information Inertia in Demand). If investors are ambiguous about the predictability of future asset values, then 15

16 (i) there is a range of bad signals over which investors do not adjust their long stock position when the unconditional risk premium is positive, (ii) there is a range of good signals over which investors do not adjust their short stock position when the unconditional risk premium is negative, (iii) information inertia is more severe when price deviates a lot from its unconditional mean, and (iv) the probability of having information inertia in demand is non monotonic in the unconditional risk premium of the asset. III Equilibrium In this section, we solve for the equilibrium stock price when a representative investors is ambiguous about the predictability of future cash flows. 18 We show that prices do not always incorporate public information that is worse than expected. Moreover, the probability of this mispricing depends on the riskiness of the stock. Suppose there is a representative investor with CARA utility who is averse to ambiguity. In equilibrium, the representative investor holds the stock and consumes the liquidating dividend d. Hence, the stock price at date 1 equals the liquidating dividend and θ = 1. The price at date 0 depends on the signal and is determined below. A Savage Benchmark The equilibrium price when the representative investor is a standard expected utility maximizer with belief ρ (in the sense of Savage (1954)) is provided in the next proposition There exists a representative investor if all investors in the economy have the same aversion to ambiguity (for a proof see Wakai (2007) or Illeditsch (2011)). Properties of the equilibrium stock price when there does not exist a representative investor are discussed in the next section. 19 The proof is straightforward and thus omitted. 16

17 Proposition 5 (Savage Benchmark). If the representative investor is a standard expected utility maximizer with belief ρ, then [ ] [ ] p(s) =E ρ d s = s γvar ρ d s = s. (15) The stock price is strictly increasing in the signal when the signal is positively correlated with the asset. This is no longer true when the representative investor is averse to ambiguity as the next theorem shows. B Ambiguity-Averse Investors Theorem 2. Let ŝ a = s γσ d σ s ρ a and ŝ b = s γσ d σ s ρ b. Then for each s there is a unique equilibrium stock price p(s). Specifically, [ ] [ ] E ρa d s = s γvar ρa d s = s if s>ŝ a [ ] [ ] p(s) = E d γvar d if ŝ b <s ŝ a (16) [ ] [ ] E ρb d s = s γvar ρb d s = s if s ŝ b. Figure 4 shows the equilibrium stock price as a function of the signal. The stock price reacts moderately to good news because in this case the worst case scenario for both the mean and the variance is a signal with a low ρ. However, if news is very bad, then investors are more worried about a low posterior mean than a high residual variance and thus the price strongly reacts to changes in the signal. There is a range of bad signal values for which the representative investor hedges against ambiguity by avoiding the signal and hence the stock price does not react to these signals. Specifically, the equilibrium stock price is constant for all signals s [ŝ b, ŝ a ]. 17

18 90 Equilibrium Price ρ a ρ b ρ = 0 [ρ a, ρ b ] Price ShatB = 77.5 ShatA = Sbar = 100 Signal Figure 4: Equilibrium Price This graph shows the equilibrium stock price as a function of the signal. The red dashed line represents an economy in which the representative investor (RI) is a Savage with belief ρ b, the blue chain-dotted line represents an economy in which the RI is a Savage with belief ρ a, and the black solid line represents an economy in which the RI is ambiguity averse ([ρ a,ρ b ]). There is a range of signals that are not priced and thus prices fail to incorporate all available public information. The parameters are d = s = 100, σ d = σ s =5,andγ =1. The left graph of Figure 5 shows the equilibrium stock price a function of the signal. The blue chain-dotted line represents an economy in which the unconditional risk premium is 0, the red dashed line represents an economy in which the unconditional risk premium is 25, and the black solid line represents an economy in which the unconditional risk premium is 50. The figure shows that the inaction region increases with the riskiness of the stock. It also shows that increasingly worse signals will not be reflected in the price if there is more risk in the economy. The right graph of Figure 5 shows that the probability of the mispricing of public information is not monotonic in the riskiness of the stock. Moreover, it also shows that conditional on increasingly bad news surprises this probability is very large for some stocks. 18

19 γ σ d 2 = 0 γ σ d 2 = 25 γ σ d 2 = 50 Price Inertia Price Inertia Two St. Dev. Bad News Suprise One St. Dev. Bad News Suprise Bad News Surprise News Surprise Price Probability Signal Risk ( γ σ 2 d ) Figure 5: Mispricing of Public Information The left graph shows the equilibrium stock price as a function of the signal for different unconditional risk premia. It shows that the information inertia in prices is more severe for risky stocks. The right graph shows the probability of having a failure of prices to incorporate public information. The black solid line shows the unconditional probability, the red chain-dotted line shows the probability conditional on bad news, the blue dashed line shows the probability conditional on having an at least one standard deviation bad news surprise, and the green dotted line shows the probability conditional on having an at least two standard deviation bad news surprise. The parameters are d = s = 100, and σ d = σ s =5. We determine the size of the inaction region and the probability of being in that region in the next proposition. 20 Proposition 6 (Failure of prices to incorporate public information). Let ŝ a = s γσ d σ s ρ a and ŝ b = s γσ d σ s ρ b. The stock price does not react to the signal for the range of signal values: ŝ a ŝ b = γσ d σ s (ρ b ρ a ). (17) The unconditional probability of the mispricing of public information is N 0,1 ( γσ d ρ a ) N 0,1 ( γσ d ρ b ). (18) The graphs in Figure 6 show the probability of having a moderate price reaction (red area), a strong price reaction (blue area), and no price reaction (black area) as a function of the riskiness of the stock. The graphs in the last row show that stocks 20 The proof follows directly from Theorem 2 and is thus omitted. 19

20 which are not very risky are more likely to react very strongly to news that are much worse than expected than stocks which are very risky. However, it is striking that stocks with moderate risk show almost no reaction to very bad ambiguous news. 100% Price Reaction 100% Price Reaction 90% 90% Unconditional Probability 80% 70% 60% 50% 40% 30% 20% 10% 0% moderate inertia strong Prob. Cond. on Bad News Surprise 80% 70% 60% 50% 40% 30% 20% 10% 0% moderate inertia strong Risk ( γ σ d 2 ) Risk ( γ σ d 2 ) Prob. Cond. on 1 St. Dev. Bad News Surprise 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% Price Reaction moderate inertia strong Prob. Cond. on 2 St. Dev. Bad News Surprise 100% 90% 80% 70% 60% 50% 40% 30% 20% 10% 0% moderate inertia strong Price Reaction Risk ( γ σ d 2 ) Risk ( γ σ d 2 ) Figure 6: Price Reaction to News All four graphs show the probability of having a moderate price reaction (red area), a strong price reaction (blue area), and no price reaction (black area) as a function of the riskiness of the stock. The first graphs shows the unconditional probability, the second graphs shows the probability conditional on bad news, the third graphs shows the probability conditional on having an at least one standard deviation bad news surprise, and the last graph shows the probability conditional on having an at least two standard deviation bad news surprise. The parameters are d = s = 100, and σ d = σ s =5. We now summarize the predictions for the equilibrium stock price. Model Predictions 3 (Failure of prices to incorporate public information). If investors are ambiguous about the predictability of future asset values, then 20

21 (i) prices do not always react to public signals that convey bad news and (ii) the probability of this mispricing depends on the riskiness of the stock. IV Heterogeneous Investors In this section, we study equilibrium prices and demands when the economy is populated by investors who may differ with respect to aversion to risk and ambiguity. We show that in equilibrium there are some investors who are inattentive to news and use their prior information when computing optimal demand. Moreover, the result that prices may not reflect all available public information in equilibrium is robust to this heterogeneity. Suppose there are H investors who all receive the same signal but may differ with respect to their initial wealth, and their aversion to risk and ambiguity. Let w 0h denote investor h s initial wealth, γ h > 0 her risk aversion coefficient, and [ρ ah,ρ bh ] her set of beliefs with ρ ah > 0 h {1,...,H}. An equilibrium in this economy is defined as follows: Definition 1 (Equilibrium). The signal-to-price map p(s) is an equilibrium s R if and only if (i) each investor chooses a portfolio θ h to maximize [ ( ) ) ] min E ρ h u h (w 0h + d p(s) θ h s = s, s R (19) ρ h [ρ ah,ρ bh ] and (ii) markets clear; i.e. H h=1 θ h =1and investors consume the liquidating d at date 1. If all investors have the same ambiguity, then we know from Wakai (2007) and Illeditsch (2011) that there exists a representative investor with risk tolerance equal to the sum of the risk tolerances of all H investors. We show in the next proposition that equilibrium prices still fail to incorporate all available public information when investors are heterogeneous in their aversion to ambiguity and their range of beliefs overlap. 21

22 Proposition 7 (Aggregation). Let 1/γ H h=1 1/γ h denote aggregate risk tolerance and let [ρ a,ρ b ] H h=1 [ρ ah,ρ bh ]. Then the equilibrium stock price is where ŝ a = s γσ d σ s ρ a and ŝ b = s γσ d σ s ρ b. p(s) = d γσ 2 d s [ŝ b, ŝ a ], (20) The size of the price inaction region is determined by the investors with the smallest ambiguity and it is increasing with aggregate risk aversion γ. Hence, the size of the inaction region and the probability of the mispricing of public information are given in equation (17) and (18) of the previous section. To gain some more intuition consider an economy in which both investors (Knights) are averse to ambiguity. The first Knight has the range of beliefs [ρ a1,ρ b1 ]=[0.2, 0.6] and the second Knight has the range of beliefs [ρ a2,ρ b2 ]=[0.4, 0.8]. The black solid line in the left graph of Figure 7 shows the equilibrium stock price as a function of the signal. For comparison, we also show the equilibrium stock price for five economies that are populated by two Savages with different beliefs. For instance, the red dotted line shows the price in an economy where the first Savage has the belief ρ 1 =0.6 and the second Savage has the belief ρ 2 =0.8. The right graph of Figure 7 shows the equilibrium demand as a function of the signal when the economy is populated by the two Knights described above. There is a range of signals that are worse than expected for which both investors use their priors when computing demands and thus the equilibrium stock price does not react to changes in these signals. 22

23 Equilibrium with Two Knights 2 Knights ([ 0.2, 0.6 ], [ 0.4, 0.8 ]) 2 Savages (0.6, 0.8) 2 Savages (0, 0.6) 2 Savages (0, 0.4) 2 Savages (0.2, 0.4) Savages with ρ = Equilibrium with Two Knights [ ρ a, ρ b ] = [ 0.2, 0.6 ] [ ρ a, ρ b ] = [ 0.4, 0.8 ] Stock Price Demand Signal Signal Figure 7: Investors with different ambiguity The left graph shows the equilibrium stock price as a function of the signal for six different economies. The black solid line represents an economy that consists of two Knights with range of beliefs ([ρ a1,ρ b1 ]=[0.2, 0.6], [ρ a2,ρ b2 ]=[0.4, 0.8]). The colored dotted lines represent economies that consist of two Savages with different beliefs. The left graph shows that if there is ambiguity, then there is a range of bad signals for which the price does not react much. The right graph shows equilibrium demand in an economy that consists of two Knights. The red dashed line shows demand of a Knight with range of beliefs [ρ a1,ρ b1 ]=[0.2, 0.6] and the black solid line shows demand of a Knight with range of beliefs [ρ a2,ρ b2 ]=[0.4, 0.8]. If the signal lies in the interval [77.92, 95.17], then at least one of the investors ignores the signal and uses her prior information when determining demand. When both investors hedge against ambiguity ([ŝ b, ŝ a ] = [85, 90]), then demand is insensitive to changes in these signals and thus equilibrium prices fail to incorporate these signals. All investors have the same risk aversion γ = 1 and the remaining parameters are d = s = 100, and σ d = σ s =5. To discuss the properties of equilibrium demand and price given in Figure 7 we consider the five different signal regions (i) (, 77.92], (ii) [77.92, 85], (iii) [85, 90], (iv) [90, 95.17], and (v) [95.17, ). Both Knights behave like Savages with beliefs ρ 1 = 0.6 andρ 2 = 0.8 for the first range of signals because if news is very bad ambiguity averse investors are more worried about a low posterior mean than a high residual variance and thus consider a high ρ. Hence, the equilibrium stock price reacts a lot to these signals. Equilibrium demand of the second Knight (red dashed line) is increasing in the signal because her worst case scenario belief (ρ =0.8) is larger than the worst case scenario belief of the second Knight (black solid line) and thus 23

24 she puts more weight on the signal. The analysis is similar for the fifth range because with good news the worst case scenario for both investors is a low ρ. For the other three ranges of signals there is at least one investor who avoids the signal and uses her prior when forming optimal demand. In other words, there is at least one investor who behaves as if there is no correlation between the signal and the dividend even though there is no ambiguity about the fact that this correlation is positive. Consider the second signal range. The first Knight sill behaves like a Savage investor with belief ρ =0.6 but the second Knight hedges against ambiguity by ignoring the signal. Hence her demand which is increasing for the first range of signals is now decreasing because neither mean nor variance depends on the signal and the equilibrium price increases with it. The equilibrium price still reacts to signals in the second region because of the first investor but not as much as for the first range of signals. Both investors ignore all signals in the third region and hence the equilibrium does not reflect these signals. The intuition for the fourth signal range is similar to the second. In this case the first investor ignores the signal when forming demand and hence in equilibrium her demand decreases with the signal. We conclude this section with a comparison of equilibrium demand for different economies. Specifically, the left graph of Figure 8 shows equilibrium demand as a function of the signal and the right graph shows it as a function of the equilibrium stock price. In both graphs the black solid line represents an economy consisting of two Knights, the red dashed line represents an economy consisting of two Savages with different beliefs, and the blue chain-dotted line represents an economy with one Knight and one Savage. The graphs show that if there is ambiguity, then equilibrium demand is neither monotone in the signal nor in the equilibrium stock price. Moreover, the left graph shows that this non monotone and erratic demand behavior only occurs for signals that are worse than expected. This is in stark contrast to an economy without ambiguity for which equilibrium demand is a smooth and monotone function of the signal If everybody has the same belief (or ambiguity) then equilibrium demand is constant. 24

25 Heterogenous Investors Heterogenous Investors Equilibrium Demand Equilibrium Demand Two Savages 0.42 One Savage and One Knight Two Knights Signal 0.4 Two Savages One Savage and One Knight Two Knights Equilibrium Stock Price Figure 8: Equilibrium Demand The left graph shows equilibrium demand as a function of the signal and the right graph shows it as a function of the equilibrium stock price. In both graphs the black solid line represents an economy consisting of two Knights with range of beliefs [ρ a1,ρ b1 ]=[0.2, 0.6] and [ρ a2,ρ b2 ]=[0.4, 0.8]). The red dashed line represents an economy consisting of two Savages with beliefs ρ 1 =0.4 andρ 2 =0.6 andthe blue chain-dotted line represents an economy with one Knight with range of beliefs [ρ a1,ρ b1 ]=[0.2, 0.8] and one Savage with belief ρ 2 =0.5. Both graphs show that equilibrium demand is non-monotone if there is ambiguity. All investors have the same risk aversion γ = 1 and the remaining parameters are d = s = 100, and σ d = σ s =5. V Conclusion We study optimal portfolios and equilibrium asset prices when there is ambiguity (uncertainty in the sense of Knight) about the predictability of future asset values. We show that this ambiguity may lead to asset demand that is insensitive to changes in news and investors use only their prior information when making portfolio decisions. In equilibrium, we show that stock prices may not react to public information that is worse than expected even though there are no information processing costs or other market frictions. 25

26 References Ahn, David, Syngjoo Choi, Douglas Gale, and Shachar Kariv, 2010, Estimating ambiguity aversion in a portfolio choice experiment, Working Paper. Backus, David K., Bryan R. Routledge, and Stanley E. Zin, 2004, Exotic preferences for macroeconomists, NBER Working Paper Bansal, Ravi, and Amir Yaron, 2004, Risks for the long run: A potential resolution of asset pricing puzzles, Journal of Finance LIX, Benigno, Pierpaolo, and Salvatore Nistico, 2012, International portfolio allocation under model uncertainty, American Economic Journal: Macroeconomics 4, Bossaerts, Peter, Paolo Ghirardato, Serena Guarnaschelli, and William Zame, 2010, Ambiguity in asset markets: Theory and experiment, Review of Financial Studies 23, Boyle, Phelim, Lorenzo Garlappi, Raman Uppal, and Tan Wang, 2012, Keynes meets markowitz: The trade-off between familiarity and diversification, Management Science 58, Campanale, Claudio, 2011, Learning, ambiguity and life-cycle portfolio allocation, Review of Economic Dynamics 14, Cao, H. Henry, Tan Wang, and Harold H. Zhang, 2005, Model uncertainty, limited market participation, and asset prices, Review of Financial Studies 18, Caskey, Judson A., 2009, Information in equity markets with ambiguity-averse investors, Review of Financial Studies 22, Chen, Hui, Nengjiu Ju, and Jianjun Miao, 2011, Dynamic asset allocation with ambiguous return predictability, Working Paper. Cochrane, John H., 2005, Asset Pricing (Princeton University Press) 1 revised edn. 26

27 Collard, Fabrice, Sujoy Mukerji, Kevin Sheppard, and Jean-Marc Tallon, 2011, Ambiguity and the historical equity premium, Working Paper. Condie, Scott, and Jayant Ganguli, 2011a, Ambiguity and rational expectations equilibria, Review of Economic Studies 78, , 2011b, Informational efficiency with ambiguous private information, Economic Theory 48, Dow, J., and S. Werlang, 1992, Uncertainty aversion, risk aversion, and the optimal choice of portfolio, Econometrica 60, Dow, James, and Sergio Ribeiro Da Costa Werlang, 1994, Nash equilibrium under knightian uncertainty breaking down backward induction, Journal of Economic Theory 64, Easley, David, and Maureen O Hara, 2009, Ambiguity and nonparticipation: The role of regulation, Review of Financial Studies 22, Ellsberg, D., 1961, Risk, ambiguity, and the savage axioms, Quarterly Journal of Economics 75, Epstein, Larry G., and Martin Schneider, 2003, Recursive multiple-priors, Journal of Economic Theory 113, 1 31., 2007, Learning under ambiguity, Review of Economic Studies 74, , 2008, Ambiguity, information quality, and asset pricing, Journal of Finance pp , 2010, Ambiguity and asset markets, Annual Review of Financial Economics 2, Epstein, Larry G., and Tan Wang, 1994, Intertemporal asset pricing under Knightian uncertainty, Econometrica 62,