observed for equities – the spread between the rate of return required for holding the market portfolio and the risk-free rate (for a discussion of the credit risk premium, see Amato and Remolona (2013)). Using the CCAPM, Grossman and Shiller (1981) were the first to show that the premium is much higher and more volatile over time than most plausible risk aversion parameters would suggest. The phenomenon was also noted by Shiller (1982) and Modigliani and Cohn (1979) but the literature truly emerged with the ground-breaking paper by Mehra and Prescott (1985).

Mehra and Prescott highlighted the difficulty that traditional models have in matching the observed excess return of stocks relative to a risk-free asset (with the degree of risk aversion viewed as realistic from a microeconomic perspective for a typical investor). The literature on the (excessive) equity risk premium in discount rates is large but inconclusive. The equity premium was subsequently refined by Epstein and Zin (1989) and Weil (1989), who developed models allowing preferences for risk and intertemporal substitution to be separated. Others explored the puzzle further, showing that one needs to distinguish between the intertemporal rate of substitution and the rate of risk aversion when trying to interpret the premium (see further Kocherlakota (1996), Mehra and Prescott (2003) and Rieger and Wang (2012)).

Higher asset price volatility can also arise if investors' risk aversion depends on macroeconomic volatility, as is the case, for example, in asset pricing models with habit formation (Campbell and Cochrane (1999)). Barro (2006) and Gabaix (2011) show that a standard model extended to allow for realistically calibrated rare-disaster probabilities can generate a high volatility of asset returns, high equity premia and low risk-free rates, which are all close to what is observed in practice.<sup>21</sup> Apart from the overreaction of asset prices to swings in cash flows, the presence of various anomalies and other indications of “mispricing” also constitute a source of puzzles (see Schwert (2003) for a review). A number of possible answers to these puzzles have been proposed. For example, in some models, the rational expectations assumption that investors optimally use current information to forecast future dividend growth is relaxed. Instead, various market failures and forms of behavioural biases are introduced, such as investors’ herding behaviour and sentiment (Barberis et al (1998)). Thaler (2005, 2015) presents a survey of the related behavioural finance literature (see also Shleifer (2000), Barberis and Thaler (2003), Barberis et al (2001) and Barberis (2013)).

Indications that investors tend to overestimate the persistence of variations in dividend growth – or, equivalently, to underprice risk – have motivated many studies (Barsky and De Long (1993)). Numerous reasons related to the functioning of markets – including limited market liquidity, “excessive” financial innovation, the perverse trading behaviour of large investors and the role of hedge funds – have also been mentioned as causes of the excessive volatility of asset prices (Bikhchandani and Sharma (2000)). Although this strand of research provides analytical models and some empirical evidence showing that asset prices are not simply determined by the present discounted value of future cash flows, it has not been able to identify definitive reasons driving the deviations from the basic models (Duffie (2010) reviews other examples of deviations).

> 21 For additional information, see also Campbell et al (2013), Chen et al (2012), Ju and Miao (2012), Albuquerque et al (2016) and Krueger et al (2016))

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