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## Determinants of Stock Market Volatility and Risk Premia

### Citations

1750 | The Equity Premium: A Puzzle
- Mehra, Prescott
- 1985
(Show Context)
Citation Context ...liefs; Rational Belief; optimism; pessimism; empirical distribution. JEL Classification Numbers: G1; G12; E43; E44; D58; D84. 1. Introduction The forces which determine equilibrium market volatility and risk premia are probably the most debated topics in the analysis of financial markets. The debate is driven, in part, by empirical evidence of market "anomalies" which have challenged students of the subject. Consumption - based asset pricing theory has had a profound impact on our view of financial markets. Early work of Leroy (1973), Lucas (1978), Breeden (1979), Grossman and Shiller (1981), Mehra and Prescott (1985), Hansen and Singleton (1983) and others show how intertemporal optimization of investor incorporates a subtle relationship between consumption growth and asset returns. However, when examined empirically, this simple relationship fails to provide a correct quantitative measure of risk 2 premia. The Equity Premium Puzzle (see, Mehra and Prescott (1985)) is a special case of the fact that assets prices are more volatile than can be explained by "fundamental" shocks. Difficult to account for risk premia are not confined to the consumption based asset pricing theory; they arise in many asset pric... |

1452 | By Force of Habit: A Consumption-Based Explanation of Aggregate Stock Market Behavior."
- Campbell, Cochrane
- 1995
(Show Context)
Citation Context ...d Lucas (1996)) hence we select . This fact has very little effect on the results.ωjt ' 3 We simulated this REE and report in Table 1 the mean and standard deviations of (i) the 3 Other approaches to the equity premium puzzle were reported by Brennan and Xia [1998], Epstein and Zin [1990], Cecchetti, Lam and Mark [1990],[1993], Heaton and Lucas [1996], Mankiw [1986], Reitz [1988], Weil [1989] and others. For more details see Kocherlakota [1996]. Some degree of excess volatility can also be explained with explicit learning mechanism which does not die out (see for example Timmermann (1996)). 4 Campbell and Cochrane (1999), (2000) assume that at habit the marginal utility of consumption and degree of risk aversion rise without bound. Hence, when consumption declines to habit, risk aversion increases, stock prices decline and risk premium rises. Although the model generates moments which are closer to those observed in the market, the theory is unsatisfactory. First, with Xt = habit, utility is . But why should the marginal utility and risk aversion explode 1 1&γ (Ct&Xt) 1&γ when Ct approaches the mean of past consumption? Campbell and Cochrane (1999, page 244) show that for the model to generate the desired mom... |

847 | Do stock prices move too much to be justified by subsequent changes in dividends?,
- Shiller, J
- 1981
(Show Context)
Citation Context ... Heterogenous beliefs; Rational Belief; optimism; pessimism; empirical distribution. JEL Classification Numbers: G1; G12; E43; E44; D58; D84. 1. Introduction The forces which determine equilibrium market volatility and risk premia are probably the most debated topics in the analysis of financial markets. The debate is driven, in part, by empirical evidence of market "anomalies" which have challenged students of the subject. Consumption - based asset pricing theory has had a profound impact on our view of financial markets. Early work of Leroy (1973), Lucas (1978), Breeden (1979), Grossman and Shiller (1981), Mehra and Prescott (1985), Hansen and Singleton (1983) and others show how intertemporal optimization of investor incorporates a subtle relationship between consumption growth and asset returns. However, when examined empirically, this simple relationship fails to provide a correct quantitative measure of risk 2 premia. The Equity Premium Puzzle (see, Mehra and Prescott (1985)) is a special case of the fact that assets prices are more volatile than can be explained by "fundamental" shocks. Difficult to account for risk premia are not confined to the consumption based asset pricing theory; th... |

811 | Asset Prices under Habit Formation and Catching Up with the Joneses. American Economic Review papers and proceedings
- Abel
- 1990
(Show Context)
Citation Context ...ar. Note that apart from the low equity premium, the REE volatility measures are lower than market data by an order of magnitude. The "Equity Premium" is not the only puzzle; the wider question is how to explain market volatility. Table 1: Simulated Moments Of Key Variables In REE (all moments are annualized) R q s σq s σR r σr ep ρRx shrp Model Data 16.71 0 .055 7.94% 3.78% 7.70% 0.79% 0.24% 0.995 0.064 Market Data 25 7.1 7% 18.00% 1.00% 5.70% 6.00% 0.100 0.333 Some have argued in favor of introducing habit formation in utility in order to generate time variability of risk aversion (e.g. see Abel (1990), (1999), Constantinides (1990), Campbell and Cochrane (1999), (2000)3). We are not persuaded by this model since habit formation can explain the equity premium only if it assumes unreasonably high degree of risk aversion4. Our alternative view proposes that risk premia are determined primarily by the structure of market expectations. Our argument is developed in sections 4, 5 and 6. First, in Section 4 we develop the general structure of equilibria where agents have time varying and diverse beliefs. These ideas apply to any 8 model with diverse beliefs, not only to a Rational Belief framework... |

778 |
Dividend yields and expected stock returns.
- Fama, French
- 1988
(Show Context)
Citation Context ...ctability of Returns and Stochastic Volatility We turn now to other dimensions of asset price dynamics, aiming to compare predictions of our theory with the empirical evidence. We study the predictability of stock returns and stochastic volatility, or GARCH, properties of stock prices and returns. Results reported were computed for a sample of 20,000 data points generated by Monte Carlo simulation of the model with a = -0.20 and b = -15.00 in Table 4. 6.1 Predictability of stock returns The problem of predictability of risky returns generated an extensive literature in empirical finance (e.g. Fama and French (1988a,1998b), Poterba and Summers (1988), Campbell and Shiller (1988), Paye and Timmermann (2003)). This debate is contrasted with the simple theoretical observation that under risk aversion asset prices and returns are not martingales, hence they contain a predictable component. It appears the disagreement is not about the empirical record but about the interpretation of the record and about the stability of the estimated forecasting models. Here we focus only on the empirical record. We examine the following: (i) Variance Ratio statistic; (ii) autocorrelation of returns and of price/dividend rat... |

677 |
Habit Formation: A Resolution of the Equity Premium Puzzle, "
- Constantinides
- 1990
(Show Context)
Citation Context ...rom the low equity premium, the REE volatility measures are lower than market data by an order of magnitude. The "Equity Premium" is not the only puzzle; the wider question is how to explain market volatility. Table 1: Simulated Moments Of Key Variables In REE (all moments are annualized) R q s σq s σR r σr ep ρRx shrp Model Data 16.71 0 .055 7.94% 3.78% 7.70% 0.79% 0.24% 0.995 0.064 Market Data 25 7.1 7% 18.00% 1.00% 5.70% 6.00% 0.100 0.333 Some have argued in favor of introducing habit formation in utility in order to generate time variability of risk aversion (e.g. see Abel (1990), (1999), Constantinides (1990), Campbell and Cochrane (1999), (2000)3). We are not persuaded by this model since habit formation can explain the equity premium only if it assumes unreasonably high degree of risk aversion4. Our alternative view proposes that risk premia are determined primarily by the structure of market expectations. Our argument is developed in sections 4, 5 and 6. First, in Section 4 we develop the general structure of equilibria where agents have time varying and diverse beliefs. These ideas apply to any 8 model with diverse beliefs, not only to a Rational Belief framework. We then explain the restricti... |

629 | Asset Prices in an Exchange Economy
- Lucas
- 1978
(Show Context)
Citation Context ...sk premium; riskless rate; over confidence; Heterogenous beliefs; Rational Belief; optimism; pessimism; empirical distribution. JEL Classification Numbers: G1; G12; E43; E44; D58; D84. 1. Introduction The forces which determine equilibrium market volatility and risk premia are probably the most debated topics in the analysis of financial markets. The debate is driven, in part, by empirical evidence of market "anomalies" which have challenged students of the subject. Consumption - based asset pricing theory has had a profound impact on our view of financial markets. Early work of Leroy (1973), Lucas (1978), Breeden (1979), Grossman and Shiller (1981), Mehra and Prescott (1985), Hansen and Singleton (1983) and others show how intertemporal optimization of investor incorporates a subtle relationship between consumption growth and asset returns. However, when examined empirically, this simple relationship fails to provide a correct quantitative measure of risk 2 premia. The Equity Premium Puzzle (see, Mehra and Prescott (1985)) is a special case of the fact that assets prices are more volatile than can be explained by "fundamental" shocks. Difficult to account for risk premia are not confined to t... |

621 |
An intertemporal asset pricing model with stochastic consumption and investment opportunities,
- Breeden
- 1979
(Show Context)
Citation Context ...skless rate; over confidence; Heterogenous beliefs; Rational Belief; optimism; pessimism; empirical distribution. JEL Classification Numbers: G1; G12; E43; E44; D58; D84. 1. Introduction The forces which determine equilibrium market volatility and risk premia are probably the most debated topics in the analysis of financial markets. The debate is driven, in part, by empirical evidence of market "anomalies" which have challenged students of the subject. Consumption - based asset pricing theory has had a profound impact on our view of financial markets. Early work of Leroy (1973), Lucas (1978), Breeden (1979), Grossman and Shiller (1981), Mehra and Prescott (1985), Hansen and Singleton (1983) and others show how intertemporal optimization of investor incorporates a subtle relationship between consumption growth and asset returns. However, when examined empirically, this simple relationship fails to provide a correct quantitative measure of risk 2 premia. The Equity Premium Puzzle (see, Mehra and Prescott (1985)) is a special case of the fact that assets prices are more volatile than can be explained by "fundamental" shocks. Difficult to account for risk premia are not confined to the consumption b... |

522 |
Permanent and temporary components of stock prices.
- Fama, French
- 1988
(Show Context)
Citation Context ...ctability of Returns and Stochastic Volatility We turn now to other dimensions of asset price dynamics, aiming to compare predictions of our theory with the empirical evidence. We study the predictability of stock returns and stochastic volatility, or GARCH, properties of stock prices and returns. Results reported were computed for a sample of 20,000 data points generated by Monte Carlo simulation of the model with a = -0.20 and b = -15.00 in Table 4. 6.1 Predictability of stock returns The problem of predictability of risky returns generated an extensive literature in empirical finance (e.g. Fama and French (1988a,1998b), Poterba and Summers (1988), Campbell and Shiller (1988), Paye and Timmermann (2003)). This debate is contrasted with the simple theoretical observation that under risk aversion asset prices and returns are not martingales, hence they contain a predictable component. It appears the disagreement is not about the empirical record but about the interpretation of the record and about the stability of the estimated forecasting models. Here we focus only on the empirical record. We examine the following: (i) Variance Ratio statistic; (ii) autocorrelation of returns and of price/dividend rat... |

505 | Yield Spreads and Interest Rate Movements: A Bird’s Eye View."
- Campbell, Shiller
- 1991
(Show Context)
Citation Context ... returns. However, when examined empirically, this simple relationship fails to provide a correct quantitative measure of risk 2 premia. The Equity Premium Puzzle (see, Mehra and Prescott (1985)) is a special case of the fact that assets prices are more volatile than can be explained by "fundamental" shocks. Difficult to account for risk premia are not confined to the consumption based asset pricing theory; they arise in many asset pricing models. Three examples will illustrate. The Expectations Hypothesis is rejected by most studies of the term structure (e.g. Backus, Gregory and Zin (1989), Campbell and Shiller (1991)) implying excessive risk premium on longer maturity debt. Foreign exchange markets exhibit a "Forward Discount Bias" (see Froot and Frankel (1989) and Engel (1996)) implying an unaccounted for risk premium for holding foreign exchange. The pricing of derivatives generates excess implied volatility of underlying securities, resulting in unaccounted for risk premia. Some treat these as unusual anomalies and give them distinct labels, presuming standard models of asset pricing can account for all normal premia. Our perspective is different. We suggest market volatility and risk premia are primar... |

452 |
Overconfidence and excess entry: an experimental approach.
- Camerer, Lovallo
- 1999
(Show Context)
Citation Context ...requires the idiosyncratic component of an agent’s belief not to be correlated with market beliefs. This is translated to the requirement that Ωz 1g j ' Cov( ρ z 1 t%1 , ρ g j t%1 ) ' 0 .Ωz 2g j ' Cov( ρ z 2 t%1 , ρ g j t%1) ' 0 Hence, anonymity restricts two components of even in a theory without restrictions on belief.Ωwg j 4.3 Modeling Tractable and Computable Functions Ψt%1(g j t ) We model so as to permit agents to be over confident by assigning to some eventsΨt%1(g j t ) higher probability than the empirical frequency. Evidence from the psychological literature (e.g. Svenson (1981), Camerer and Lovallo (1999) and references) shows agents exhibit such behavior. In 15 some cases this behavior may be irrational but this is not generally true. Institutional and technical changes are central to an economy and past statistics do not provide the best forecasts for the future. Deviations from empirical frequencies reflect views based on limited recent data about changed conditions. Agents have financial incentive to make such judgments since major financial gains are available to those who bet on the correct market changes. We now discuss the asymmetries built into modeling .Ψt%1(g j t ) Asymmetry and Int... |

445 |
Stochastic Consumption, Risk Aversion and Temporal Behavior of Asset Returns,"
- SINGLETON
- 1983
(Show Context)
Citation Context ...imism; pessimism; empirical distribution. JEL Classification Numbers: G1; G12; E43; E44; D58; D84. 1. Introduction The forces which determine equilibrium market volatility and risk premia are probably the most debated topics in the analysis of financial markets. The debate is driven, in part, by empirical evidence of market "anomalies" which have challenged students of the subject. Consumption - based asset pricing theory has had a profound impact on our view of financial markets. Early work of Leroy (1973), Lucas (1978), Breeden (1979), Grossman and Shiller (1981), Mehra and Prescott (1985), Hansen and Singleton (1983) and others show how intertemporal optimization of investor incorporates a subtle relationship between consumption growth and asset returns. However, when examined empirically, this simple relationship fails to provide a correct quantitative measure of risk 2 premia. The Equity Premium Puzzle (see, Mehra and Prescott (1985)) is a special case of the fact that assets prices are more volatile than can be explained by "fundamental" shocks. Difficult to account for risk premia are not confined to the consumption based asset pricing theory; they arise in many asset pricing models. Three examples wi... |

400 | The forward discount anomaly and the risk premium: a survey of recent evidence,
- ENGEL
- 1996
(Show Context)
Citation Context ...Prescott (1985)) is a special case of the fact that assets prices are more volatile than can be explained by "fundamental" shocks. Difficult to account for risk premia are not confined to the consumption based asset pricing theory; they arise in many asset pricing models. Three examples will illustrate. The Expectations Hypothesis is rejected by most studies of the term structure (e.g. Backus, Gregory and Zin (1989), Campbell and Shiller (1991)) implying excessive risk premium on longer maturity debt. Foreign exchange markets exhibit a "Forward Discount Bias" (see Froot and Frankel (1989) and Engel (1996)) implying an unaccounted for risk premium for holding foreign exchange. The pricing of derivatives generates excess implied volatility of underlying securities, resulting in unaccounted for risk premia. Some treat these as unusual anomalies and give them distinct labels, presuming standard models of asset pricing can account for all normal premia. Our perspective is different. We suggest market volatility and risk premia are primarily determined by the structure of agents’ expectations called "market state of belief." Diversity and dynamics of beliefs are then the root cause of price volatili... |

383 |
Evaluating the Effects of Incomplete Markets on Risk Sharing and Asset Pricing
- Heaton, Lucas
- 1996
(Show Context)
Citation Context ... σx ' 0.03256 Ours is a theoretical paper aiming to draw qualitative conclusions. We use realistic parameter values since we wish our simulations to result in numerical values which are close to the observed data. For simplicity we assume , a constant hence total resources, or GNP, equal (1+2T)Dt and theω j t ' ω Dividend/GNP ratio is . As we model only income from publically traded stocks, the1 1 % 2ω Dividend/(Household Income) ratio should include corporate dividends but exclude self employed income and imputed income from other asset categories. This ratio is about 15% (see survey data in Heaton and Lucas (1996)) hence we select . This fact has very little effect on the results.ωjt ' 3 We simulated this REE and report in Table 1 the mean and standard deviations of (i) the 3 Other approaches to the equity premium puzzle were reported by Brennan and Xia [1998], Epstein and Zin [1990], Cecchetti, Lam and Mark [1990],[1993], Heaton and Lucas [1996], Mankiw [1986], Reitz [1988], Weil [1989] and others. For more details see Kocherlakota [1996]. Some degree of excess volatility can also be explained with explicit learning mechanism which does not die out (see for example Timmermann (1996)). 4 Campbell and C... |

358 |
The Equity Premium Puzzle and the Risk-free Rate Puzzle
- Weil
- 1989
(Show Context)
Citation Context ... 0.28 19.12 4.38 9.27% 20.53% 0.66% 5.21% 8.62% 0.20 0.42 25.00 7.10 7.00% 18.00% 1.00% 5.70% 6.00% 0.10 0.33 Table 2 shows the riskless rate declines towards 1% simply because the RBE becomes more volatile. This affects both the volatility of individual consumption growth rates as well as their correlation with the growth rate of aggregate consumption. In an REE this correlation is close 21 to 1 but not in an RBE where variability of individual consumption growth depends upon the agents’ beliefs. The low observed riskless rate has been a central issue in the equity premium puzzle debate (see Weil (1989)) and the effect of the parameter b goes to the heart of this issue. Our theory offers the intuitive explanation that non normal belief densities with fat tails and high intensity propagate high market volatility hence risk, making financial safety costly. Now compare Table 2 with the empirical record. As volatility increases and the riskless rate r reaches 0.66% we see that (i) qs declines below 25, (ii) R rises above 7.0%, (iii) rises aboveσR 18.0%, (iv) the Sharp Ratio rises above 0.33. Some reflection shows that these are reasonable conclusions for a symmetric volatile economy in which bea... |

349 | Stock prices, earnings, and expected dividends’, The
- Campbell, Shiller
- 1988
(Show Context)
Citation Context ...to other dimensions of asset price dynamics, aiming to compare predictions of our theory with the empirical evidence. We study the predictability of stock returns and stochastic volatility, or GARCH, properties of stock prices and returns. Results reported were computed for a sample of 20,000 data points generated by Monte Carlo simulation of the model with a = -0.20 and b = -15.00 in Table 4. 6.1 Predictability of stock returns The problem of predictability of risky returns generated an extensive literature in empirical finance (e.g. Fama and French (1988a,1998b), Poterba and Summers (1988), Campbell and Shiller (1988), Paye and Timmermann (2003)). This debate is contrasted with the simple theoretical observation that under risk aversion asset prices and returns are not martingales, hence they contain a predictable component. It appears the disagreement is not about the empirical record but about the interpretation of the record and about the stability of the estimated forecasting models. Here we focus only on the empirical record. We examine the following: (i) Variance Ratio statistic; (ii) autocorrelation of returns and of price/dividend ratios; (iii) regressions of cumulative returns, and (iv) the predic... |

319 | Mean Reversion in Stock Prices: Evidence and Implications,"
- Poterba, Summers
- 1988
(Show Context)
Citation Context ...stic Volatility We turn now to other dimensions of asset price dynamics, aiming to compare predictions of our theory with the empirical evidence. We study the predictability of stock returns and stochastic volatility, or GARCH, properties of stock prices and returns. Results reported were computed for a sample of 20,000 data points generated by Monte Carlo simulation of the model with a = -0.20 and b = -15.00 in Table 4. 6.1 Predictability of stock returns The problem of predictability of risky returns generated an extensive literature in empirical finance (e.g. Fama and French (1988a,1998b), Poterba and Summers (1988), Campbell and Shiller (1988), Paye and Timmermann (2003)). This debate is contrasted with the simple theoretical observation that under risk aversion asset prices and returns are not martingales, hence they contain a predictable component. It appears the disagreement is not about the empirical record but about the interpretation of the record and about the stability of the estimated forecasting models. Here we focus only on the empirical record. We examine the following: (i) Variance Ratio statistic; (ii) autocorrelation of returns and of price/dividend ratios; (iii) regressions of cumulative... |

286 |
Are We All Less Risky and More Skillful Than Our Fellow Drivers?”
- Svenson
- 1981
(Show Context)
Citation Context ... that anonymity requires the idiosyncratic component of an agent’s belief not to be correlated with market beliefs. This is translated to the requirement that Ωz 1g j ' Cov( ρ z 1 t%1 , ρ g j t%1 ) ' 0 .Ωz 2g j ' Cov( ρ z 2 t%1 , ρ g j t%1) ' 0 Hence, anonymity restricts two components of even in a theory without restrictions on belief.Ωwg j 4.3 Modeling Tractable and Computable Functions Ψt%1(g j t ) We model so as to permit agents to be over confident by assigning to some eventsΨt%1(g j t ) higher probability than the empirical frequency. Evidence from the psychological literature (e.g. Svenson (1981), Camerer and Lovallo (1999) and references) shows agents exhibit such behavior. In 15 some cases this behavior may be irrational but this is not generally true. Institutional and technical changes are central to an economy and past statistics do not provide the best forecasts for the future. Deviations from empirical frequencies reflect views based on limited recent data about changed conditions. Agents have financial incentive to make such judgments since major financial gains are available to those who bet on the correct market changes. We now discuss the asymmetries built into modeling .Ψt... |

233 | Risk Premia and Term Premia in General Equilibrium. - Abel - 1999 |

229 |
Predictability of stock returns: Robustness and economic significance,”
- Pesaran, Timmermann
- 1995
(Show Context)
Citation Context ...to exhibit persistence. Market volatility is then time dependent. It changes with the market state of beliefs and hence it has a predictable component as in (19)-(20). These results extend the earlier and similar result in Kurz and Motolese (2001). The virtue of the above argument is that it explains stochastic volatility as an endogenous consequence of equilibrium dynamics. Some “fundamental” shocks (i.e. an oil shock) surely cause market volatility, but it has been empirically established that market volatility cannot be explained consistently by repeated “fundamental” exogenous shocks (see Pesaran and Timmermann (1995)). Our explanation of stochastic volatility is thus consistent with the empirical evidence 7. Concluding Remarks This paper presents a unified paradigm which proposes that market volatility is driven primarily by market expectations. This conclusion extends our previous work (cited earlier) on the subject. The central new development is the formal introduction of the state of belief as a key tool of General Equilibrium analysis and a corresponding requirement for agents to forecast the market state of belief. Agents forecasting the state of belief of “others” is a precise mathematical structur... |

210 |
Does the Stock Market Rationally Reflect Fundamental Values?,
- Summers
- 1986
(Show Context)
Citation Context ...eatures of the empirical evidence and we report it in Table 8. Table 8: The behavior of the regression slopes in (18) Time horizon Model Empirical Record k χk R 2 χk R 2 1 2 3 4 5.03 8.66 11.16 13.10 0.08 0.14 0.18 0.21 5.32 9.08 11.73 13.44 0.07 0.11 0.15 0.17 To conclude the discussion of predictability, we observe that the empirical evidence reported by Fama and French (1998a, 1998b), Campbell and Shiller (1988), Poterba and Summers (1998) and others is consistent with asset price theories in which time-varying expected returns generate predictable, mean-reverting components of prices (see Summers (1986)). The important question left unresolved by these papers is what drives the predictability of returns implied by such mean-reverting components of prices? Part of the answer is the persistence of the dividend growth rate via the equilibrium map (6). Our theory offers a second and stronger persistent mechanism which is also seen in (6). It shows these results are primarily driven by the dynamics of market state of beliefs which exhibit correlation across agents and persistence over time. Agents go through bull and bear states causing their perception of risk to change and expected returns to v... |

184 | Social Value of Public Information.
- Morris, Shin
- 2002
(Show Context)
Citation Context ... circumstances at a date t that call for a deviation at t from the benchmark. In short, is a description of how the model of agent j deviates from the statistical forecast implied by (2). g jt In this paper we assume that at any date the state of belief is a realization of a process of the form (5) .g jt%1 ' λzg j t % λ zj x (xt & x () % ρg j t%1 , ρ g j t%1 - N( 0 , σ 2 g j ) Persistent states of belief which depends upon current market data fit different cases of economies with diverse beliefs. We consider three examples to illustrate how one may think about them. 6 See Woodford (2003) , Morris and Shin (2002), Allen Morris and Shin (2003) and others 10 (i) Measure of Animal Spirit. “Animal Spirit” expresses intensity at which agents carry out investments and this, in turn, is based upon expected rewards. identifies the probability an agentg jt assigns to high or low rates of return hence can be interpreted as a measure of “animal spirit.”g jt (ii) Learning Unknown Parameters. In a learning context agents use prior distributions on unknown parameters. Since defines an agent’s belief about the profit growth process we can identify asg jt g j t a posterior parameter of its mean value function. A post... |

181 | Imperfect Common Knowledge and the Effects of Monetary Policy.
- Woodford
- 2003
(Show Context)
Citation Context ...g an evaluation of circumstances at a date t that call for a deviation at t from the benchmark. In short, is a description of how the model of agent j deviates from the statistical forecast implied by (2). g jt In this paper we assume that at any date the state of belief is a realization of a process of the form (5) .g jt%1 ' λzg j t % λ zj x (xt & x () % ρg j t%1 , ρ g j t%1 - N( 0 , σ 2 g j ) Persistent states of belief which depends upon current market data fit different cases of economies with diverse beliefs. We consider three examples to illustrate how one may think about them. 6 See Woodford (2003) , Morris and Shin (2002), Allen Morris and Shin (2003) and others 10 (i) Measure of Animal Spirit. “Animal Spirit” expresses intensity at which agents carry out investments and this, in turn, is based upon expected rewards. identifies the probability an agentg jt assigns to high or low rates of return hence can be interpreted as a measure of “animal spirit.”g jt (ii) Learning Unknown Parameters. In a learning context agents use prior distributions on unknown parameters. Since defines an agent’s belief about the profit growth process we can identify asg jt g j t a posterior parameter of its me... |

159 | Existence of equilibrium of plans, prices and price expectations in a sequence of markets”,
- Radner
- 1972
(Show Context)
Citation Context ... t%1) The introduction of individual and market states of belief has two central implications: (A) The economy has an expanded state space, including market belief . in thiszt zt' ( z 1 t , z 2 t )0ú 2 paper. Hence diverse beliefs create new uncertainty which is the uncertainty of what others may do. This adds a component of volatility which cannot be explained by “fundamental” shocks. Denoting usual state variables by , the price process { , t = 1, 2, ...} is defined by a map likest (q s t , q b t ) (6) . qst qbt ' Ξ( st , z 1 t , z 2 t , . . . , z N t ) Our equilibrium is thus an incomplete Radner (1972) equilibrium with an expanded state space. (B) To forecast prices agents must forecast market beliefs. Although all use (6) to forecast prices, agents’ forecasts are different since each forecasts given his own state . This is a(st%1 , zt%1) g j t feature of the Keynes Beauty Contest: to forecast equilibrium prices you must forecast beliefs of other agents. A Beauty Contest does not entail higher order of beliefs: at t you form belief about market belief but the date t+1 market belief is not a probability about your date t belief state7. zt%1 We now return to the economy with two agent types a... |

155 |
Forward discount bias: Is it an exchange risk premium?”
- Froot, Frankel
- 1989
(Show Context)
Citation Context ...emium Puzzle (see, Mehra and Prescott (1985)) is a special case of the fact that assets prices are more volatile than can be explained by "fundamental" shocks. Difficult to account for risk premia are not confined to the consumption based asset pricing theory; they arise in many asset pricing models. Three examples will illustrate. The Expectations Hypothesis is rejected by most studies of the term structure (e.g. Backus, Gregory and Zin (1989), Campbell and Shiller (1991)) implying excessive risk premium on longer maturity debt. Foreign exchange markets exhibit a "Forward Discount Bias" (see Froot and Frankel (1989) and Engel (1996)) implying an unaccounted for risk premium for holding foreign exchange. The pricing of derivatives generates excess implied volatility of underlying securities, resulting in unaccounted for risk premia. Some treat these as unusual anomalies and give them distinct labels, presuming standard models of asset pricing can account for all normal premia. Our perspective is different. We suggest market volatility and risk premia are primarily determined by the structure of agents’ expectations called "market state of belief." Diversity and dynamics of beliefs are then the root cause ... |

154 |
Asset prices, consumption, and the business cycle",
- Campbell
- 1998
(Show Context)
Citation Context ...4c) θ1t % θ 2 t ' 1 (4d) .b 1t % b 2 t ' 0 3. A Rational Expectations Equilibrium (REE) Strictly speaking we cannot evaluate the REE since the true output process has not been specified. We thus define an REE to be the economy in which all agents believe that (2) is the true output process. To evaluate the volatility of this REE in an annual model we specify parameters of (2). Unfortunately, different estimates of the parameter values are available, depending upon time span of data, unit of time (annual vs. quarterly) and definition of terms (see, for a sample, Backus, Gregory and Zin (1989), Campbell (2000) and Rodriguez (2002) Appendix 2, based on Shiller in (http://www.econ.yale.edu/shiller)). We use the annual estimates in Campbell (2000) Table 3, which are consistent with Mehra and Prescott (1984). Hence, for the rest of this paper we set , β'0.96 , and , all within the empirically estimated range. γ'2.00 , x ('0.01773 λx ' &0.117 σx ' 0.03256 Ours is a theoretical paper aiming to draw qualitative conclusions. We use realistic parameter values since we wish our simulations to result in numerical values which are close to the observed data. For simplicity we assume , a constant hence total re... |

149 | The Equity Premium and the Concentration of Aggregate Shocks,” - Mankiw - 1986 |

132 |
The determinants of the variability of stock market prices,
- Grossman, Shiller
- 1981
(Show Context)
Citation Context ...r confidence; Heterogenous beliefs; Rational Belief; optimism; pessimism; empirical distribution. JEL Classification Numbers: G1; G12; E43; E44; D58; D84. 1. Introduction The forces which determine equilibrium market volatility and risk premia are probably the most debated topics in the analysis of financial markets. The debate is driven, in part, by empirical evidence of market "anomalies" which have challenged students of the subject. Consumption - based asset pricing theory has had a profound impact on our view of financial markets. Early work of Leroy (1973), Lucas (1978), Breeden (1979), Grossman and Shiller (1981), Mehra and Prescott (1985), Hansen and Singleton (1983) and others show how intertemporal optimization of investor incorporates a subtle relationship between consumption growth and asset returns. However, when examined empirically, this simple relationship fails to provide a correct quantitative measure of risk 2 premia. The Equity Premium Puzzle (see, Mehra and Prescott (1985)) is a special case of the fact that assets prices are more volatile than can be explained by "fundamental" shocks. Difficult to account for risk premia are not confined to the consumption based asset pricing theory; th... |

121 | Prospect Theory and Asset Prices,
- Barberis, Huang, et al.
- 2001
(Show Context)
Citation Context ...on of the kind documented by Poterba and Summers (1988), Fama and French (1998a) and Campbell and 28 Shiller (1988). Thus, apart from the very short returns which exhibit positive autocorrelation, the model reproduces the empirical record reasonably well. In Table 7 we report the autocorrelation function of the price-dividend ratio. The table shows the model generates a highly autocorrelated price/dividend ratio which matches reasonably well the behavior observed in the U.S. stock market data. The empirical record in Tables 6 and 7 is for NYSE data covering the period 1926-1995 as reported in Barberis et al. (2001). Table 7: Autocorrelation of Price-Dividend Ratio Model Empirical Recordcorr(q st ,q s t&i) i = 1 i = 2 i = 3 i = 4 i = 5 0.695 0.485 0.336 0.232 0.149 0.700 0.500 0.450 0.430 0.400 6.1.3 Mean Reversion of Log-Returns Mean reversion of stock returns was studied under several models. We compare our results with the results of Fama and French (1988a). Thus, consider the regression model (17) .kkt%k ' αk % δkk k t % υt ,k The evidence suggests that stock prices have a random-walk and a stationary component (see Fama and French (1988a)), depending upon the horizon k. If there was no stationary co... |

110 | First-order” risk aversion and the equity premium puzzle, - Epstein, Zin - 1990 |

105 | Mean reversion in equilibrium asset prices, - Cecchetti, Lam, et al. - 1990 |

95 | Explaining the Poor Performance of Consumption-Based Asset Pricing Models.”
- Campbell, Cochrane
- 2000
(Show Context)
Citation Context ...at t;B jt - stock price at date t;qst - the discount price of a one period bond at t;qbt - non capital income of agent j at date t;Λjt Ht - information at t, recording the history of all observables up to t. Given probability belief , agent j selects portfolio and consumption plans to solve the problemQ jt (3a) Max (C j,θj ,B j) EQ j [j 4 t '0 βt 1 1 & γ ( C jt ) 1 & γ |Ht ] subject to: (3b) .C jt % q s t θ j t % q b t B j t ' Λ j t % ( q s t % Dt )θ j t&1 % B j t&1 We assume additively separable, power utility over consumption, a model that failed to generate premia in other studies (see, Campbell and Cochrane (2000)). We focus on diverse beliefs hence 6 assume the two utility functions are the same. Introduce the normalization for j = 1, 2 .ωjt / Λ j t Dt , c jt / C jt Dt , q st / qst Dt , b jt / B jt Dt With this normalization the budget constraint becomes (3b’) .c jt % q s t θ j t % q b t b j t ' ω j t % (q s t % 1)θ j t&1 % b j t&1 e &xt The Euler equations are (4a) (c jt ) &γq st ' βEQ jt [ ( c j t%1 ) &γ (1 % q st%1 )e (1 & γ )xt%1 |Ht ] (4b) (c jt ) &γq bt ' βEQ jt [ ( c j t%1 ) &γ e &γxt%1 |Ht ] and the market clearing conditions are then (4c) θ1t % θ 2 t ' 1 (4d) .b 1t % b 2 t ' 0 3. A Rational ... |

71 |
A structural model for stock return volatility and trading volume.
- Brock, LeBaron
- 1996
(Show Context)
Citation Context ...s are primarily driven by the dynamics of market state of beliefs which exhibit correlation across agents and persistence over time. Agents go through bull and bear states causing their perception of risk to change and expected returns to vary over time. Equilibrium asset prices depend upon states of belief which then exhibit memory and mean reversion. Hence both prices and returns exhibit these same properties. 6.2 GARCH Behavior of the Price-dividend Ratio and of the Risky Returns Stochastic volatility in asset prices and returns is well documented (e.g. Bollerslev, Engle and Nelson (1994), Brock and LeBaron (1996)). In partial equilibrium finance it is virtually standard to model asset prices by stochastic differential equations, assuming an exogenously driven stochastic volatility. But where does stochastic volatility come from? Dividends certainly do not exhibit stochastic volatility. One of the most important implication of our theory is that it explains 30 why asset prices and returns exhibit stochastic volatility. We start by presenting in Figures 5 and 6 the results of simulated 500 observations: in Figure 5 we report price/dividend ratios and in Figure 6 the associated risky rates of return. The... |

62 | The Equity Premium and the Risk-Free Rate: Matching the Moments”, - Cecchetti, Lam, et al. - 1993 |

59 |
Excess volatility and predictability of stock prices in autoregressive dividend models with learning,
- Timmermann
- 1996
(Show Context)
Citation Context ...rvey data in Heaton and Lucas (1996)) hence we select . This fact has very little effect on the results.ωjt ' 3 We simulated this REE and report in Table 1 the mean and standard deviations of (i) the 3 Other approaches to the equity premium puzzle were reported by Brennan and Xia [1998], Epstein and Zin [1990], Cecchetti, Lam and Mark [1990],[1993], Heaton and Lucas [1996], Mankiw [1986], Reitz [1988], Weil [1989] and others. For more details see Kocherlakota [1996]. Some degree of excess volatility can also be explained with explicit learning mechanism which does not die out (see for example Timmermann (1996)). 4 Campbell and Cochrane (1999), (2000) assume that at habit the marginal utility of consumption and degree of risk aversion rise without bound. Hence, when consumption declines to habit, risk aversion increases, stock prices decline and risk premium rises. Although the model generates moments which are closer to those observed in the market, the theory is unsatisfactory. First, with Xt = habit, utility is . But why should the marginal utility and risk aversion explode 1 1&γ (Ct&Xt) 1&γ when Ct approaches the mean of past consumption? Campbell and Cochrane (1999, page 244) show that for the ... |

57 | Instability of Return Prediction Models,"
- Paye, Timmerman
- 2006
(Show Context)
Citation Context ...price dynamics, aiming to compare predictions of our theory with the empirical evidence. We study the predictability of stock returns and stochastic volatility, or GARCH, properties of stock prices and returns. Results reported were computed for a sample of 20,000 data points generated by Monte Carlo simulation of the model with a = -0.20 and b = -15.00 in Table 4. 6.1 Predictability of stock returns The problem of predictability of risky returns generated an extensive literature in empirical finance (e.g. Fama and French (1988a,1998b), Poterba and Summers (1988), Campbell and Shiller (1988), Paye and Timmermann (2003)). This debate is contrasted with the simple theoretical observation that under risk aversion asset prices and returns are not martingales, hence they contain a predictable component. It appears the disagreement is not about the empirical record but about the interpretation of the record and about the stability of the estimated forecasting models. Here we focus only on the empirical record. We examine the following: (i) Variance Ratio statistic; (ii) autocorrelation of returns and of price/dividend ratios; (iii) regressions of cumulative returns, and (iv) the predictive power of the 27 dividen... |

48 |
Endogenous Uncertainty and Market Volatility."
- Kurz, Motolese
- 2001
(Show Context)
Citation Context ...tool we use to describe the beliefs of agents is the “market state of belief.” It is explained in details in this paper. The RB rationality is compatible with several known theories. An REE is a special case of an RBE. Most models of Bayesian learning satisfy the RB rationality principle. Also, several models of Behavioral Economics satisfy this principle for some parameter values. Earlier papers using the RBE rationality principle have also argued that agents’ beliefs are central to explaining market volatility (e.g. Kurz (1996), (1997a), Kurz and Schneider (1996), Kurz and Beltratti (1997), Kurz and Motolese (2001), Kurz and Wu (1996) and Nielsen (1996)). These 4 papers aimed to explain a list of financial "anomalies." (for a unified treatment see Kurz and Motolese (2001)). The RBE theory was used by Kurz (1997b) and Nielsen (2003) to explain the volatility of foreign exchange rates. These papers used OLG models where exogenous shocks are discrete and agents have a finite set of belief states. Wu and Guo (2003) (2004) study speculation and trading in the steady state of an infinite horizon model. This paper’s contribution consists of four parts: (i) ours is an infinite horizon model, (ii) all random var... |

43 | Risk Aversion and the Martingale Property of Stock Returns”. - LeRoy - 1973 |

38 |
On the structure and diversity of rational beliefs.
- Kurz
- 1994
(Show Context)
Citation Context ...y an economy with stock and riskless bond markets and formulate a financial equilibrium model with diverse and time varying beliefs. Agents’ states of belief play a key role in the market, requiring an endogenous expansion of the state space. To forecast prices agents must forecast market states of belief which are beliefs of “others” hence our equilibrium embodies the Keynes “Beauty Contest.” A “market state of belief” is a vector which uniquely identifies the distribution of conditional probabilities of agents. Restricting beliefs to satisfy the rationality principle of Rational Belief (see Kurz (1994), (1997)) our economy replicates well the empirical record of the (i) moments of the price/dividend ratio, risky stock return, riskless interest rate and the equity premium; (ii) Sharp ratio and the correlation between risky returns and consumption growth; (iii) predictability of stock returns and price/dividend ratio as expressed by: (I) Variance Ratio statistic for long lags, (II) autocorrelation of these variables, and (III) mean reversion of the risky returns and the predictive power of the price/dividend ratio. Also, our model explains the presence of stochastic volatility in asset prices... |

27 |
Rational Belief Structures and Rational Belief Equilibria.
- Nielsen
- 1996
(Show Context)
Citation Context ...the “market state of belief.” It is explained in details in this paper. The RB rationality is compatible with several known theories. An REE is a special case of an RBE. Most models of Bayesian learning satisfy the RB rationality principle. Also, several models of Behavioral Economics satisfy this principle for some parameter values. Earlier papers using the RBE rationality principle have also argued that agents’ beliefs are central to explaining market volatility (e.g. Kurz (1996), (1997a), Kurz and Schneider (1996), Kurz and Beltratti (1997), Kurz and Motolese (2001), Kurz and Wu (1996) and Nielsen (1996)). These 4 papers aimed to explain a list of financial "anomalies." (for a unified treatment see Kurz and Motolese (2001)). The RBE theory was used by Kurz (1997b) and Nielsen (2003) to explain the volatility of foreign exchange rates. These papers used OLG models where exogenous shocks are discrete and agents have a finite set of belief states. Wu and Guo (2003) (2004) study speculation and trading in the steady state of an infinite horizon model. This paper’s contribution consists of four parts: (i) ours is an infinite horizon model, (ii) all random variables are continuous, (iii) AR(1) proc... |

25 | The Role of Expectations in Economic Fluctuations and the Efficacy of Monetary Policy." - Kurz, Jin, et al. - 2005 |

21 |
Asset Price Volatility and Trading Volume with Rational
- Wu, Guo
- 2004
(Show Context)
Citation Context ...BE rationality principle have also argued that agents’ beliefs are central to explaining market volatility (e.g. Kurz (1996), (1997a), Kurz and Schneider (1996), Kurz and Beltratti (1997), Kurz and Motolese (2001), Kurz and Wu (1996) and Nielsen (1996)). These 4 papers aimed to explain a list of financial "anomalies." (for a unified treatment see Kurz and Motolese (2001)). The RBE theory was used by Kurz (1997b) and Nielsen (2003) to explain the volatility of foreign exchange rates. These papers used OLG models where exogenous shocks are discrete and agents have a finite set of belief states. Wu and Guo (2003) (2004) study speculation and trading in the steady state of an infinite horizon model. This paper’s contribution consists of four parts: (i) ours is an infinite horizon model, (ii) all random variables are continuous, (iii) AR(1) processes describe beliefs and exogenous shocks, and (iv) we explicitly model agents’ beliefs about the market state of belief, which are beliefs about the beliefs of others. We argue that this is the crucial property needed for understanding the volatility and risk premia in financial markets. The Main Results . First, "Belief states" are developed as a tool for equ... |

20 | Endogenous uncertainty in a general equilibrium model with price contingent contracts.
- Kurz, Wu
- 1996
(Show Context)
Citation Context ...e beliefs of agents is the “market state of belief.” It is explained in details in this paper. The RB rationality is compatible with several known theories. An REE is a special case of an RBE. Most models of Bayesian learning satisfy the RB rationality principle. Also, several models of Behavioral Economics satisfy this principle for some parameter values. Earlier papers using the RBE rationality principle have also argued that agents’ beliefs are central to explaining market volatility (e.g. Kurz (1996), (1997a), Kurz and Schneider (1996), Kurz and Beltratti (1997), Kurz and Motolese (2001), Kurz and Wu (1996) and Nielsen (1996)). These 4 papers aimed to explain a list of financial "anomalies." (for a unified treatment see Kurz and Motolese (2001)). The RBE theory was used by Kurz (1997b) and Nielsen (2003) to explain the volatility of foreign exchange rates. These papers used OLG models where exogenous shocks are discrete and agents have a finite set of belief states. Wu and Guo (2003) (2004) study speculation and trading in the steady state of an infinite horizon model. This paper’s contribution consists of four parts: (i) ours is an infinite horizon model, (ii) all random variables are continuou... |

12 |
Floating Exchange Rates vs. A Monetary Union Under Rational Beliefs:
- Nielsen
- 2003
(Show Context)
Citation Context ...of Bayesian learning satisfy the RB rationality principle. Also, several models of Behavioral Economics satisfy this principle for some parameter values. Earlier papers using the RBE rationality principle have also argued that agents’ beliefs are central to explaining market volatility (e.g. Kurz (1996), (1997a), Kurz and Schneider (1996), Kurz and Beltratti (1997), Kurz and Motolese (2001), Kurz and Wu (1996) and Nielsen (1996)). These 4 papers aimed to explain a list of financial "anomalies." (for a unified treatment see Kurz and Motolese (2001)). The RBE theory was used by Kurz (1997b) and Nielsen (2003) to explain the volatility of foreign exchange rates. These papers used OLG models where exogenous shocks are discrete and agents have a finite set of belief states. Wu and Guo (2003) (2004) study speculation and trading in the steady state of an infinite horizon model. This paper’s contribution consists of four parts: (i) ours is an infinite horizon model, (ii) all random variables are continuous, (iii) AR(1) processes describe beliefs and exogenous shocks, and (iv) we explicitly model agents’ beliefs about the market state of belief, which are beliefs about the beliefs of others. We argue th... |

12 | The Equity Premium: A Solution. - Reitz - 1988 |

12 | Timing: Better Than Buy-and-Hold Strategy. - Shilling - 1992 |

11 |
Coordination and Correlation
- Kurz, Schneider
- 1996
(Show Context)
Citation Context ...ust have forecast functions which vary over time. The tool we use to describe the beliefs of agents is the “market state of belief.” It is explained in details in this paper. The RB rationality is compatible with several known theories. An REE is a special case of an RBE. Most models of Bayesian learning satisfy the RB rationality principle. Also, several models of Behavioral Economics satisfy this principle for some parameter values. Earlier papers using the RBE rationality principle have also argued that agents’ beliefs are central to explaining market volatility (e.g. Kurz (1996), (1997a), Kurz and Schneider (1996), Kurz and Beltratti (1997), Kurz and Motolese (2001), Kurz and Wu (1996) and Nielsen (1996)). These 4 papers aimed to explain a list of financial "anomalies." (for a unified treatment see Kurz and Motolese (2001)). The RBE theory was used by Kurz (1997b) and Nielsen (2003) to explain the volatility of foreign exchange rates. These papers used OLG models where exogenous shocks are discrete and agents have a finite set of belief states. Wu and Guo (2003) (2004) study speculation and trading in the steady state of an infinite horizon model. This paper’s contribution consists of four parts: (i) o... |

6 |
Evolving Term Structure with Heterogenous Beliefs.
- Fan
- 2003
(Show Context)
Citation Context ... t%1 In any application one assumes the parameters of (7a)-(7c) are known by all. With regard to x, we have set values for in Section 2. We normalize by setting variances of equal to 1. (λx , x ( , σx) ρ z j t%1 To specify parameters of the zj equations recall that z measures how optimistic agents are about future returns on investment. With this in mind we used forecasts reported by the Blue Chip Economic Indicators and the Survey of Professional Forecasters, and “purged” them of observables. We then estimated principal components to handle multitude of forecasted variables (for details, see Fan (2003)). The extracted indexes of beliefs imply regression coefficients around 0.5 - 0.8 hence we set . Investors’ forecasts of financial variables such as corporate profits, are highlyλz 1'λz 2'0.7 correlated and a value of is realistic. In this paper we study only symmetric economiesσz 1z 2 ' 0.90 where agents differ only in their beliefs hence we assume . The evidence shows thatλz 1 x ' λ z 2 x ' λ z x positive profit shocks lead agents to revise upward their economic growth forecasts implying . λzx > 0 Our best guess of this parameter leads us to set but we discuss it again later. λzx ' 0.9 To w... |

4 |
Consumption, the Persistence of Shocks, and Asset Pricing Puzzles.
- Rodriguez
- 2002
(Show Context)
Citation Context ...(4d) .b 1t % b 2 t ' 0 3. A Rational Expectations Equilibrium (REE) Strictly speaking we cannot evaluate the REE since the true output process has not been specified. We thus define an REE to be the economy in which all agents believe that (2) is the true output process. To evaluate the volatility of this REE in an annual model we specify parameters of (2). Unfortunately, different estimates of the parameter values are available, depending upon time span of data, unit of time (annual vs. quarterly) and definition of terms (see, for a sample, Backus, Gregory and Zin (1989), Campbell (2000) and Rodriguez (2002) Appendix 2, based on Shiller in (http://www.econ.yale.edu/shiller)). We use the annual estimates in Campbell (2000) Table 3, which are consistent with Mehra and Prescott (1984). Hence, for the rest of this paper we set , β'0.96 , and , all within the empirically estimated range. γ'2.00 , x ('0.01773 λx ' &0.117 σx ' 0.03256 Ours is a theoretical paper aiming to draw qualitative conclusions. We use realistic parameter values since we wish our simulations to result in numerical values which are close to the observed data. For simplicity we assume , a constant hence total resources, or GNP, equa... |

1 | Risk Premiums in the Term Structure: Evidence from Artificial Economies. - paper - 2003 |