Salient theory and asset pricing for risk averse agents
The Model for Risk Averse Investors. Analysis of equilibrium price for risk averse agents. Risk aversion and wealth effects for salient and non-salient cases. Comparison of risk loving and risk averse equilibrium prices. The Model for Risk Loving Agents.
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Федеральное государственное автономное образовательное учреждение высшего профессионального образования
Национальный исследовательский университет «Высшая школа экономики»
Международный институт экономики и финансов
Направление подготовки 38.03.01 «Экономика»
Образовательная «Программа двух дипломов по экономике НИУ ВШЭ и Лондонского университета»
Теория «заметных доходов» и их ценообразование при нерасположенных к риску инвесторах
Salient theory and asset pricing for risk averse agents
Выполнила: Степанова Елизавета
Научный руководитель PhD, доцент,
Макаров Дмитрий Сергеевич
The objective of this work is to determine the influence caused by presence of salient dividend payments in case of risk averse investors. The earlier research states that salience causes mispricing of assets equilibrium price compared to their fundamental value and is a reason of growth-value anomaly in the behavioral finance. This work analyzes the stated hypothesis of mispricing for different types of risk preferences, compares the magnitude of price mismatch for risk averse and risk loving agents. The effects of risk aversion coefficient and wealth on equilibrium price are also considered. In the last chapter of the work the private case of time discounting is provided for risk neutral agents decisions under salience condition.
1. The Model for Risk Averse Investors
1.1 The economic setting
1.2 Analysis of equilibrium price for risk averse agents
1.3 Risk aversion and wealth effects for salient and non-salient cases
2. The Model for Risk Loving Agents
2.1 Comparison of risk loving and risk averse equilibrium prices
3. Salient Asset Pricing Model for Risk Neutral Agents and Time Discounting factor
List of References
The proposed in this work model is founded on the concept of salient dividend payments found by Bordalo, Gennaioli and Shleifer (2013). In the paper authors explain particular variation in asset prices under uncertainty with the presence of salience. This theory suggests that agents put more weight on salient payoffs. Taylor and Thompson (1982) state that: “Salience refers to the phenomenon that when one's attention is differentially directed to one portion of the environment rather than to others, the information contained in that portion will receive disproportionate weighting in subsequent judgments.” Taylor, Shelley E.; Thompson, Suzanne C. “Stalking the elusive "vividness" effect”..Psychological Review, Vol 89(2), Mar 1982, 155-181. Salient assets are characterized by relatively high returns compared to the average return on the market. In the paper Bordalo, Gennaioli and Shleifer (2013) explain how the mispricing of assets arises in presence of salience, show the variation in investment decision theory and bring up a point on Allais Paradoxes1 and preferences of the agents. The general concept of salience was proposed in the earlier paper Bordalo, Gennaioli and Shleifer (2010) according to which salience arises as an investment decision choice under uncertainty in case of lotteries a payoff. The origin of salience comes from the Prospect Theory Daniel Kahneman and Amos Tversky (1979)., its relation with salience will be discussed further in the first part of my work.
In my work I would like to explain the mispricing of assets by expanding the model proposed by Bordalo, Gennaioli and Shleifer (2013) Further mentioned as BGS(2013) by modifying it for risk averse and risk loving agents and show the difference in results for all three types of agents, interpret wealth and risk aversion coefficient effects on the equilibrium price and add a discount factor for risk neutral agents in the model. The salience in my model is determined by the weight that agents assign to the payoffs, when agents assign higher or lower weight comparing to an average payoff. Large dividend payments have higher weight in the equilibrium price determination while low dividend payments have smaller weight. The obtained result for risk averse agents who pay more attention to salient payoffs leads to the fact of overpricing large positive returns relative to average return and underpricing of large negative returns compared to average market return.
I found that for all three types of agents, exposed to risk, salient payoff influence the equilibrium price and the magnitude of mispricing varies according to variation in sign of risk aversion coefficient and change in agents' preferences to risk. The risk aversion of agents is determined by their willing to minimize risk from taking an investment opportunity and interest in receiving risk premium. While risk lovers are interested in receiving higher return and chose the most risky investment decision. So the model helps to explain risk-switching motive from risk averse to risk loving investors, that was found by Allais (1953), who showed that the risk-aversion coefficient varies for two lotteries. According to Bordalo, Gennaiolli and Shleifer (2010) agents are risk loving when their attention is drawn to highest lottery payoffs and risk averse when they are concerned about lowest payoffs.
BGS (2013) suggested that the expansion of their assets pricing model for risk neutral agents and discounting factor is good point to examine. There exists two time periods in the model for investment decisions and agents can deposit additional amount of income from the first period, hence I propose a private case when risk neutral agent can use this opportunity and receive additional gain in the second period with some time discount. As a result the model modification shows an intuitive result of discounting factor being present in the equilibrium price with the same salient payoff as in simple model for risk neutral agents.
The key findings that I came across in my work are connected with the wealth and risk aversion coefficient influence on the equilibrium price of the observed assets. The variation in risk aversion of individuals does not lead to the exclusion of salience presence in the model. The magnitude of mispricing between rationally stated price and price under salient payoffs depends on the value of risk aversion coefficient. For risk averse agents the difference between rational and salient price will be smaller comparing to risk loving agents, intuitively risk lovers are concerned about higher returns at higher risk, hence large price of an asset does not frighten them. The main finding states that the pattern of mispricing is present for all types of agents and in case of expansion of the model for risk neutral agents who can deposit the saved in the first period amount the salience is also present for the equilibrium outcomes and mismatch between rational and salient price is observed.
In my work I would like to focus on the concept of mispricing in assets as it gives rise to such problem on financial market as arbitrage gains, equity puzzling evidence as Mehra and Prescott (1985) suggest. Another reason is growth-value puzzle as growth stocks have higher price because of large salient positive increases, while value stocks are exposed to such events as bankruptcy, hence represent salient downsides. As my model accounts for agents with different types of risk, the Allais (1953) paradox is examined due to switch of risk preferences of agents. Hence, the main objective of this work is to show that mispricing of salient assets is observed for all types of agents and the level of risk exposure influences the equilibrium price and the magnitude of mismatch between equilibrium price under salient assumption and rationally stated equilibrium price.
The structure of the work is as follows: firstly, I would like to modify the model for Salient Theory in two periods for risk averse agents, hence solving the BGS (2013) expected utility for risk averse agents and observe the equilibrium price that will be compared with equilibrium price for risk neutral agents. Moreover, the effects of initial wealth and risk aversion on equilibrium price will be examined in the second chapter of my work. In the third chapter the equilibrium price will be analyzed under different ranges of risk aversion coefficient and equilibrium price for risk lovers will be compared to the proposed in chapter 1 model. The last part of the model is concerned with adding a discount factor into the model for risk neutral agents and compares it with no time discounting case equilibrium price. In the next chapter the model for changes in risk avers coefficient will be produced to show the variability in risk premium and show different outcomes for case suggested in the chapter 1 of the work produces for risk-averse agents. And lastly the conclusion about salient model of asset pricing for risk averse agents and time discounting for risk neutral agents will be made, the main propose of the conclusion is to show the difference or similarities between risk neutral and risk averse models and see how time discounting factor affects the equilibrium model for risk neutral investors.
Chapter 1. The Model for Risk Averse Investors
1.1 The Economic Setting
In the first part of my work I would like to propose a model of salient assets prices for risk-averse investors, give the general determination of Salience Theory and determine why the model produce interesting results for case of risk averse agents. I would like to specify the concept of Salience Theory, its origin and than explain the model produced by BGS (2013) that is modified in this thesis.
The main idea of Salience Theory refers to the fact that agents weight the possible dividend payment before they make an investment decision and according to this weight, they choose the optimal equilibrium amount of j-th asset to purchase, that is represented as in this assets pricing model. Hence by this parameter the optimal quantity and price on the market can be determined. The innovation that I would like to propose is the presence of risk averse investors in the model, where investors prefer certain outcomes to uncertainty and have a fixed amount of relative risk aversion to be protected from risk due to presence of uncertainty in the model.
Salience occurs in the model when one can observe large difference between observed payoff, in this model dividend payment, and the average payoff, in this model is determined by average dividend payment across all states of the world. It was first introduce in Bordalo, Gennaioli and Shleifer (2010) «Salience Theory of Choice Under Risk» and assumed that the larger the difference between average and observed payoff the more salient is an asset. Agents draw more attention to more salient payoffs and assume they are less risky due to some behavioral anomaly present on the market, for example irrational trading behavior, when agents consider only the salient payoffs and draw large weight on such a return in their portfolio, hence the properties of rational diversification will be neglected. The way agents weight the payoffs depend on the rank that they assign to each asset. The higher the rank the more is the difference between the observed return and the average on the market. This concept is discussed more in the setting of the proposed in the thesis model.Salience suggests that under uncertainty some returns exceed market average return and can lead to wrong investment decisions or arbitrage opportunities. Market participants consider these so called salient payoffs more important and predictable than other assets.
In my model it is assumed that agents' investment choice is determined by the maximization of expected utility over two consumptions (c0, c1) in two periods (t0, t1). The set-up of the model is as follows according to BGS (2013) paper on salience and assets prices: there are two time periods t0 and t1, at the first period risk averse agents receive an endowment of wealth - , and purchase some amount of risky asset, we assume that agents purchase j-th asset from j = 1…J possible observed assets Agents receive a dividend payment for investing in the j-th asset at t1. The uncertainty of dividend payoffs is present in the model as they can vary according to some external factors, hence there are states of the world: where any state of the world belongs to the set: Any state of the world can occur with the same probability the probabilities for all states are equal to the assumptions of the model. The sum of all probabilities across states is 1. The dividend payment is determined as - income that agents get from investing in j-th asset in the observed state of the world s. Hence the state of the world is known at t1.
Salience of the payoff can be spotted by comparing the observed dividend payment - , with an average payoff across all states of the world - . Where an average payment is the total number of assets available on the market divided by the sum of all dividend payments across all states.
Bordalo, Gannioli and Shleifer (2010) show that at first to each dividend payment rank of salience is assigned, the rank depends on the difference between observed dividend payment and the average payment for the market, the larger this difference the more salient is the payoff. Second, based on ranking of the payment the specific weight is assigned to each dividend from J possible cases. Hence, the salient payment is a payoff on a dividend above or below the averaged dividend return. The larger the difference between average return and the observed j-th dividend the more salient it is. In the model salient payoffs are ranked according to their dividend gains, so the rank is calculated for each asset and lays in between , where asset with rank 1 is the most salient, and the one with rank S is the least salient. Then according to the produced rank the weight to each asset is assigned. It is crucial to determine the weight of each asset as it helps to determine the level of salience of payoffs, as investors assign larger weights for more salient returns.
Where is the degree that captures how investors ignore non-salient (normal) payoffs. So when this degree is equal to one the rational investor case occurs, as the weight of an asset will be equal to one over its probability of occurrence. In the model we are interested in the case when as this states that investors overweight certain payoffs, hence the salience adds to the model.
I would like to include a risk-aversion coefficient to modify the existing model of Bordalo, Gennaioli and Shleifer (2013). The relative risk aversion coefficient is presented in the utility of each individual investor. The phenomenon of risk aversion was proposed by Arrow (1971) and Pratt (1964). Risk aversion implies that agents prefer certain outcomes with a particular expected value to uncertain outcomes with the same expected value. Hence agents avoid uncertain decisions with high risk this leads to the marginally diminishing utility function. Arrow (1971) and Pratt (1964) analyzed the twice-differentiable utility function, where the risk aversion coefficient is determined as the proportion between the second and the first derivatives.
There are several standard models for maximizing utility for risk averse investors, which correspond to the proposed properties of risk aversion. The choice of the model depends on risk aversion coefficient that can be constant absolute or constant relative. There are several more utility functions for risk averse agents as DARA, but I would like to talk about CARA and CRRA and explain why in my model CRRA utility function is used.
Constant absolute coefficient can state that agents have a fixed amount as a risk premium.
Where is some constant number and represents the marginal substitution between second and first derivative.
For the constant relative risk aversion coefficient agents have some fixed share of wealth, which is a compensation for risk.
Hence CRRA utility produces the same coefficient as CARA but multiplied by the level of initial wealth.
The standard utility functions which can be used to obtain the given above results are:
For my work CRRA function is used, as agents hold fixed share in premium for risk for any given amount of wealth.
In case of the uncertain situation with equal dividend payments our investor prefers more certain payoff to more uncertain, hence he will try to reduce risk exposure. According to Rabin (2000), the presence of risk averse investors implies a concave utility function, so the utility function is marginally diminishing of wealth and helps to explain such pattern in human behavior as disliking uncertainty in life-time investment choice. Moreover, the risk averse investors have a fixed elasticity of substitution between c0 and c1.
The CRRA utility is defined as a function of -consumption over period 0 and 1, i=0,1; and relative risk aversion coefficient.
The properties of CRRA utility state that the relative risk aversion coefficient is constant and equal to , as it is derived in equation (3).
Now we move to the determination of the consumption bundles over two periods. In the first period agents receive endowment in the form of initial wealth and purchase of an asset. The initial wealth level is represented by W and cannot exceed the future expected dividend return by the model assumption. The amount of purchased assets is defined by , where j states for each asset from J amount observed. The consumption in the first period t=0 is given by equation (4) where states for price of each asset j from J assets observed. Hence the expression shows the consumption bundle of asset for any individual.
In the next period individuals receive dividend payoffs, which differ according to the state of the world. Hence the consumption in the second period, t=1, is given by equation (5), where the weight assign to any j-th asset in any states of the world s. The agents observed endowment in period t=1 in the form of income on asset j () and some additional income from saving or selling activities which is assumed to be 1 for simplification of the model.
Combining the utility for risk averse agent in the equation (2) with the consumption sets given by (4) and (5) we can obtain the utility in both periods for a risk maximizing agent. The salience will be present in this model in the form of uncertainty in the second period and weight, determined by equation (1), assigned to each asset. Then the utility that investor maximize in period 0 and 1 is represented as:
To derive the optimal value of amount of asset purchased and the price of asset we need to maximize the expected utility for an agent. According to the theory of the expected utility each period's utility should be multiplied by its probability of occurrence. The first period utility will happen for sure, hence the probability is 1, for the second period consumption there are S states of the world which can occur each with the equal probability, so the expected utility then will be determined by
The expected utility theory states that the expected benefit from decision-making is equal to the product of each contingent commodities and its corresponding probability. Expected utility is assumed to be a linear function according to von Neumann-Morgenstern (1944).
According to von Neumann-Morgenstern(1944) expected utility for risk averse agents is assumed to be is represented by equation (8) and is subject to the given consumption bundles. Individuals can choose amount of asset purchased , hence I suggest the expected utility should be maximized by the amount of asset purchased. I consider a case for risk aversion coefficient being more than 1, RRA.
Hence the utility function to maximize is represented by equation (9) and the first order condition is derived in the appendix №1.
The left side of the equation (9) corresponds to the consumption in the first period: individual observes the endowment wealth W and spends his sources on purchasing the amount of investment. The probability of consumption in the first period is 1, as there is no uncertainty in this period.
The right part of the model shows the expected outcome for all S states of the word: the individual gains more than she has invested and weights it according to the given function for weighting salient payoffs.
By maximizing the given expected utility by the amount of asset purchased the first order condition is determined in the appendix №1 and represented by equation (10).
The equilibrium is determined by the optimal buying decision that is determined by amount traded derived by maximizing equation (9), and market equilibrium occurs at , according to BGS (2013). Hence the model for risk averse investors is concerned with the price traded at the market equilibrium, I assume , plugging it in the FOC and deriving the optimal market price.
Equation (11) states that the equilibrium market price consists of the expected dividend payment and its covariance with the weight assigned to each payoff. The further analysis of the equilibrium price is represented in the next section of this chapter.
1.2 Analysis of Equilibrium Price for Risk Averse Agents
The equilibrium market price, derived in equation (11) has the same feature as was produced for risk-neutral agents by BGS (2013), which states that the equilibrium price is equal to the expected dividend payment and covariance of dividend with the weight assigned to it. Intuitively risk aversion of agents has influenced the equilibrium price, as individual adjusts the expected dividend payments and its relation with the weight assigned to investors' initial wealth W in the first period and to the relative risk aversion coefficient .
The observed price shows that there exists some mispricing on the market according to the presence of salient payoffs in the model. Which means that assets are not treated fairly, which will occur in case of equal weights assigned to each j-th asset, hence the relation between the weight of the dividend payoff and dividend payment will be zero. As a result in case of fair pricing the equilibrium price for risk-averse agent would be equal to its adjusted to initial wealth and relative risk aversion coefficient expected dividend payment. The comparison between non-salient payoff, rational market set-up, and salient payoffs is provided below to show the difference for risk-averse agents and compare it with the risk-neutral case stated by BGS (2013).
At first, I would like to show what the equilibrium outcome is when equal weights are observed, weight that agents assign for each payoff form dividend payment.
Suppose the degree of neglecting non-salient payoffs , then the weight for each individual asset is , for all observed dividend payoffs, this corresponds to the rational investor case, as all assets are treated equally on the market, plugging in in equation (1) and substituting it into (11), we observe , the weight of an asset does not depend on its dividend payoffs, as investors consider that all assets are the same.
In this case the equilibrium price is represented by the equation (12).
The prices observed for the rational risk averse investor state that it is the expected dividend payoff multiplied by expected payoff relative to initial wealth and adjusted for risk averse agent. If we consider the case of , the risk neutral price derived by BGS is observed and is equal to the expected payoff of a dividend.
Now we move to the place where salience is represented in the market as some stocks influence investment decisions more than other, due to their salient payoffs. So the weight is determined as it was stated in the set-up of the model in equation (1). The equilibrium price of an asset determined by equation (11), if the premium is required for any investor on the market, the sign of the covariance of can vary across states of the world S. It is interpreted that negative covariance implies lowest payoffs are the salient ones, while positive covariance means highest payoffs are the salient ones according to BGS (2010). Hence when the relation is negative, in the risk-neutral case the investors focus on the downside potential of the stock, as the equilibrium price will go down, hence the pattern of underweighting of an asset occurs. In the proposed model for risk-averse investors the negative relation between the way investors weight an asset and its dividend payment also lead to underpricing of an asset for any . Underpricing implies that agents underweight the assets payoffs hence their prices are lower than the rational economy case will produce, suggested by the equation (10). In case when the highest payoffs are the most salient the overpricing occurs for both risk-neutral and risk-averse investors for any .
Comparing the case of risk-neutral and risk-averse investors in both cases the mispricing will be present due to the presence of salient payoffs. As a result the hypothesis that salience leads to overpricing of most salient and high payoffs and underpricing of most salient low payoffs is proved. As the equation (11) and (12) state the case for salient payoffs being presented in the economy lead to higher or lower price on an asset, because the salient low payoffs imply negative covariance of dividend payment and the weight that agent assign to it, as agents are afraid of purchasing a low dividend payment and reduce the weight of this asset in a portfolio. While the positive dividend payoff state that agents value it more, hence the weight and dividend payoff have positive relation, so there exists positive covariance between the weight that agent assign to the dividend payment and its expected amount, as a result larger price on an asset will be presented on the market according to equation (11).
The obtained results lead to the same conclusions about mispricing being present in the model according to salient payoffs as the BGS (2013) model states and shows that the level of risk aversion or initial wealth can influence the magnitude of this mispricing. The larger the relative risk aversion coefficient, the lower will be the equilibrium price, as agents try to reduce risk due to uncertainty being present in the model and agents value uncertain risky assets at lower prices.
The initial wealth in equation (11) states that the higher its level, the lower will be the equilibrium price for any relation between dividend payments and weight of the salient payoff. Hence this also supports the case of risk averse agents as the higher initial endowment they have, the less risky behavior they choose to protect their income.
As it was stated above the equilibrium price represented by equation (11) is dependent on the relative risk aversion coefficient and initial wealth, weights assigned to payoffs on the dividends and degree at which investors neglect the non-salient payoffs. I would like to examine how the equilibrium price can change according to variations in initial wealth and risk aversion coefficient in the next chapter.
1.3 Comparison Risk Aversion Coefficient Effect and Wealth Effect for Salient and Non-salient cases
From the obtained equilibrium price of j-th asset we can see what risk aversion or initial wealth can bring to the equilibrium price. Let's consider a derivative of price of any j-th asset with respect to change in risk aversion coefficient.
Equation (13) is analyzed more specifically in appendix №2. For positive dividend payments , , as it is assumed that for positive returns agents rank them as “good” returns and want to have more of this positive return for their investment purchase. The same logic applies for negative returns, which are assumed to be bad for agents,
so , ,
as agents want to use negative return payment less than the average return payments.
The observed equation shows that the relation between equilibrium price and relative risk aversion coefficient depends on sign and magnitude of the covariance between dividend payoffs and weight assigned by non-rational agents, as the model is conducted under salient theory assumptions of non-equal weights assigned to the observed assets. By examining equation (13) we can states that there is positive relation between changes in the equilibrium price and relative risk aversion in case of positive covariance between the weight of the payoff and the positive dividend return.
In case of strong negative relation between dividend payments and the way agents weight assets the effect can be considered to be negative, as the expected return on a dividend will be also negative in this case. The complete mathematical analysis of equation (13) is provided in the appendix №2.
This relation shows that the effect of increase in risk aversion has ambiguous effect on asset price depending on the sing of the expected return of the dividend. If the dividend promises positive payment than the price of this dividend will increase in case of more risk averse investors. In case of negative return the price should decrease when agents get too risk averse.
If there is no salience present in the economy, covariance of weight of dividend payoffs and dividends is zero, as dividend payments are treated equally. Plugging in equation (1) we get that the weight assigned to each payoff is the same, hence the covariance will be zero according to BGS model of salient payoffs. The derivative in this case will be represented by equation (14):
In no salience case the change in price is positively correlated with the change in risk aversion, which is consistent with the previous research of Pratt (1964), which is true for the positive expected dividend payments. So when agents' relative risk aversion coefficient increases the price of an asset should also follow positive change in relative risk aversion and increase. This can be intuitively explained, as agents require more risk compensation for higher . According to Pratt (1964) the premium for risk is increasing with the level of risk in the model. Risk premium is defined as level at which the agent is indifferent between receiving a risk and some sure amount that is insurance. Where the return on asset is defined as the difference between this period price and last period price separated by previous period price. Hence the larger the current period price the larger the return on asset and risk premium.
It is observed that Salience leads to contradictions in relation between risk aversion of agents and the equilibrium price on the market. If the degree of neglecting non-salient payoffs is less than 1, meaning , the agents should receive additional risk premium, as , which states that the lowest payoffs are most salient one. In this case the relation between price and risk aversion can be negative. Hence when the agents require more premium for risk the price of the asset can go down. However, this can be possible only if the expected dividend payoff is less than the covariance between its weight and payoff. In case of large assets being the most salient one the covariance will be positive and the price of the asset will increase with the increase in risk aversion of the investor.
As a result we can conclude that for any relative risk aversion coefficient the largest payoffs being the salient ones lead to positive relation between equilibrium price and risk aversion of an agent, representing the same result as the rational investor case, but the magnitude of a change for irrational agent varies with the risk aversion coefficient value. The lowest payoffs being the most salient ones lead to possible negative relation between price of an asset and relative risk aversion coefficient, hence this argues with the Pratt (1964) result of risk premium.
Now we consider the relation between initial wealth that investors receive in the first period and the equilibrium price that is determined in the next period.
The observed in (15) result shows that there is negative relation between equilibrium market price and initial wealth endowment for any positive relative risk aversion coefficient in case of salience being present in the model. It is also assumed that the negative relation is present for expected payoff on a dividend being above covariance between dividend payoff and the weight that agents assign to a payoff. In this case the more the agent receives as an initial wealth the lower equilibrium price can be expected on the market. Now we consider the case when covariance between weight of payoffs and dividends is zero as agents are rational and assign equal weight to each payoff and obtain equation (16).
The equation (16) is consistent with non-salient payoffs and states that the sign of the model for any level of risk aversion coefficient above zero the wealth effect is negative. As a result the higher the wealth an agent receives in the first period the lower equilibrium price will be observed on the assets market. Obtaining (16) it can be concluded that the higher the dividend payoff the higher the magnitude of negative relation between wealth effect and equilibrium price, it can be also seen that the larger the risk aversion coefficient the larger the change in price.
Given equation (15) and (16) we can conclude that initial wealth effect on price of asset has negative effect that does not depends on such factor as the relation between expected dividend stream and covariance between dividend stream and weight of a payoff and the relation between weigh of the payoff and dividend payment for salient case of asset pricing.
For the non-salient case the relation is also always negative, as the change in price negatively depends on the amount of expected dividend payment and relative risk aversion coefficient. For the wealth effect with presence of salience in the model even larger uncertainty on equilibrium price is observed due to changes in initial wealth. As in equation (16) the wealth effect depends only on the risk aversion coefficient and dividend payment that are known to investor, hence, the future movement in prices can be predicted. As Barberis, Huang, Santos (2001) suggest in the Prospect Theory that the increase in initial wealth leads to more risk averse agents, in the analyzed model for salient payoffs and risk averse investors increase in initial wealth brings decrease in equilibrium prices, which can be interpreted as precautionary behavior of agents, as they are concerned with minimizing their risks and underestimate assets prices due to presence of salience and wealth effect on their decision.
The conclusion that can be drawn from expected utility maximization for risk averse agent, represented by equation (9), and obtaining an optimal price for any j asset is dependent on the expected dividend payoff of an asset, its covariance with weight that any agent assign to the payoff, initial wealth of an asset and relative risk aversion coefficient. It could be also stated that salient payoffs lead to deviation of equilibrium price from rational case equilibrium price, hence overpricing of large payoffs and underpricing of small payoffs relative to average dividend payoff is observed. This proves that for risk averse agents the hypothesis of mispricing is consistent with BGS model for risk neutral agents.
This section examines the how equilibrium price reacts to changes in risk aversion coefficient, as according to Kahnemn and Tversky (1979) risk aversion is varying according to the size of stake of a gamble, hence it is important to understand the expected change in price due to risk aversion variation. The analyzed model showed that there exists an ambiguous effect on price due to presence of salient payoffs in the financial market.
Salience of low payoffs leads to negative relation between price of an asset and the level of risk aversion of investors. The logic is as follows: when agents observe the negative difference between observed dividend and the average value they decrease their demand on this dividend and substitute risky asset for risk free. Hence the risk aversion of agents leads them to decrease the optimal price for an asset.
While in case of large positive payoffs being salient the relation between price and risk aversion coefficient is positive. This could be explained as the larger the positive difference between average and observed return the more salient it is for agents, hence they focus their attention on purchasing this asset, that can increase the demand for this asset. So even when agents are risk averse and their exposure to risk increase they still are willing to purchase the asset at higher price. So this case contradicts with the risk aversion theory, but is consistent with another agent's risk preferences, which will be analyzed in the second part of this chapter.
investor equilibrium price risk
Chapter 2. The Model for Risk Loving Agents
2.1 Comparison of Risk Loving and Risk Averse Equilibrium Prices
Now I would like to consider a particular case of risk aversion coefficient variation. Due to its origin raised from the Prospect Theory by Kahneman and Tversky (1979), who derived a model which describes decision making under risk as a further development of utility theory which was not able to capture pattern of overweighting several payoffs and underweight the one with lower probability of occurrence. I would like to pay attention to the fact that Prospect Theory leads to the isolation effect, which corresponds to the inconsistency of agents preferences, hence agents vary their risk aversion coefficient according to the uncertainty in the investment climate. Which is analyzed in this chapter as the equilibrium prices under salience for risk loving and risk averse agents are compared.
In my work I assume that investors compare the gain with the average value, which was represented in the first chapter of the thesis. Hence, the hypothesis that the risk aversion coefficient is varying according to the past performance of stock returns in the assets pricing model is derived and proved in this chapter.
As Arrow-Pratt theory states the risk aversion coefficient can determine the premium for risk as , where is represented in the first chapter as risk aversion coefficient, V(a) is a variance of some examined amount a. The risk premium is defined as the sure amount of money that would make a decision maker indifferent between receiving the risky return "a" and receiving the non-random amount .
The behavior of risk averse agent, as it was mentioned before, is determined as requiring higher risk premium for more uncertain situation or receiving a positive risk premium for a given investment choice. The risk neutral agent is indifferent between choosing risky or risk free asset, hence his risk premium is equal to zero and requires zero risk aversion coefficient. While risk lover is an investor who requires additional risk for making an investment decision for specific return. Hence for the risk lover the premium for risk is below zero, which origins from negative risk aversion coefficient and the convex utility function corresponds to this investment behavior.
Considering an equilibrium price stated on the market under salient payoff and risk loving agent, we obtain equation (11), where the price is determined by the expected dividend payoffs, their covariance with the assigned weight and level of risk aversion for risk loving agent. Hence the equation will be modified by the negative nature of risk aversion coefficient.
Comparing the case of risk averse agent and risk lover, hence (11) and (17), for constant level of and and , the obtained price for risk lover will exceed price stated by risk averse agent in the presence of salient payoffs in the economy. Moreover, if we consider case for rational investor, who weight all the assets the same, according to the suggested by equation (12) the same difference will be present for the two types of agents, as risk lovers require higher price for given amount of assets and uncertainty level. The complete analysis is derived in appendix №3. This statement is consistent with Pratt (1964) that risk averse agents require lower risk and return on investment in comparison with risk lovers.
Analyzing the derivative effect of increase of risk aversion coefficient on equilibrium price for risk loving agent leads, again as for risk averse agents, to the ambiguous effect according to equation (13). The final result depends on the tradeoff between expected dividend payments, on the sing and magnitude of its covariance with weight produced for salient payoffs by the equation (1). The observed effect of change in relative risk aversion coefficient will depend on the dividend payoff being low and salient or high and salient. As for low and salient payoffs their weight for a portfolio will be lower than the rational case expects, and the covariance will be negative. It should be also mentioned that for risk averse agent we analyze the sign of equation (18), and for risk loving agent, when , to come up with the expected change in price due to change in risk aversion coefficient, generally determined by equation (13).
The equation (18) above shows a part of derivate of price with respect to risk aversion coefficient for the case of . Hence the agents here are risk averse and if the negative covariance of weight and dividend payoffs is larger than the expected dividend payment, for any odd power represented by , the derivative will be negative. Hence risk averse, who are irrational and weight low payoff lower than perfect competitive financial market suggests, can expect decrease in the equilibrium price due to increase in the risk aversion coefficient.
For the case of large payoffs being the salient one, the covariance represented in the equation above will be positive, hence the whole derivative in (13) shows a positive relation between equilibrium price and change risk aversion coefficient for any level of above zero.
If we consider negative risk aversion coefficient, the equation (18) will have the same sign properties as for risk averse investors: when , the change in risk aversion will correspond to the opposite changes in equilibrium price for odd power, given by , hence should be odd for negative relation between price of an asset and risk aversion coefficient to hold. In the case of positive covariance, the relation between variables is considered to be positive. So for the risk loving agents the absolute value of risk aversion coefficient and equilibrium price move in the same direction, given positive payoffs being the most salient ones. This case also illustrates that the overpricing and underpricing will be present for all types of investors.
It is observed that for the change in risk aversion will bring higher change in price than in case of , because mathematically it could be represented by comparison of two cases for equation (18), where negative corresponds to higher numerator of the derivative and hence the larger derivative itself.
Risk loving investors were found to be more sensitive to changes in risk aversion, as the magnitude of the change in price is larger for risk lovers than for risk averse agents. This can be observed by examining the power of equation (18), as it represents the main difference between equilibrium outcomes for risk averse and risk loving investors. As a result the second part of chapter 2 states that salience affects the magnitude of change in prices due to changes in risk aversion for all types of agents, expect risk neutral. Moreover, salience leads to higher mispricing in case of risk lovers due to larger value of the derivative, represented by equation (18) and assuming risk aversion coefficient below zero.
The hypothesis of time varying risk aversion coefficient is also proved. When the dividend payment is low the relation between risk aversion coefficient and equilibrium is negative, which leads to agents being risk averse as they decrease their demand on asset due to lower payments on it. For the case of large payments, the agents are considered o be risk loving, as when their risk aversion coefficient grows they are willing to make investment purchase at higher price. This conclusion is consistent with Bordalo, Gennaiolli and Shleifer (2010) that agents are risk loving when they focus on highest payoffs and risk averse when they draw attention to lowest payoffs.
The discussed finding on Salience Theory being present in the model is consistent for both risk averse and risk loving investors. As it was examined the negative risk aversion coefficient does not play role for the sign of the derivative represented by equation (13). Risk loving investors were found to be more sensitive to changes in risk aversion, as the magnitude of the change in price is larger for risk lovers than for risk averse agents. This can be observed by examining the power of equation (18), as it represents the main difference between equilibrium outcomes for risk averse and risk loving investors. As a result the second part of chapter two states that salience affects the magnitude of change in prices due to changes in risk aversion for all types of agents, expect risk neutral. Moreover, salience leads to higher mispricing in case of risk lovers due to larger value of the derivative, represented by equation (18) and assuming risk aversion coefficient below zero.
Chapter 3. Salient Asset Pricing Model for Risk Neutral Agents and Time Discounting factor
The next modification of the model states that agents can borrow and lend unlimited amount at some interest rate . But it should be noted that the provided here model is a particular case of model represented by Bordalo, Gennaioli and Shleifer (2013). The main assumption of the model is that agents are risk neutral and invest in dividend in period t=o, they receive a dividend payment and some additional amount that could be obtained by selling. More specific set-up of the model will be provided further in this section. I assume that agents are risk-neutral for model simplification and to compare the case represented by BGS (2013) and the equilibrium price with time discounting factor being present in the model.
I would like to specify why I chose to represent one more modification of the salient assets pricing model. Time discounting factor is one of the main determinants for agents who make investment decisions during more than one time period. The nature of discounted utility model was spotted by Fisher (1930), represented as a decision between two consumptions in different periods. The interest rate there played a role of discounting factor for the consumption in the next period. Fisher (1930) derived agents' time preferences considering diminishing marginal utility function. Fisher explained the patterns in time preferences according to the initial level of wealth or risk. These concepts will be also considered in this part of the word. Frederick, Loewenstein and O'Donoghue (2002) critically analyzed the time varying presences of investors according to the presence of discounting factor. According to their paper the intertemporal consumption choice is based on decision between investment and consumption in two time periods.
Discounting utility model observe some anomalies, such as hyperbolic discounting or the fact that discounting factor can vary according to the size of an observed gain or loss or according to a time period. Such variability in discounting factor during time was found by Thaler (1981). Benzion, Rapoport and Yagil (1989) and Thaler (1981) reported that gains are discounted more than losses, which was called as a “sign effect”. Another effect, “magnitude”, was found by Thaler (1981) and Marjorie K. Shelley (1993) and states that the outcome size influence the determination of the discounting factor, hence for larger amounts corresponds larger discount factor.
Moreover, another existing anomaly of change in discounting factor states that it varies according to the level of uncertainty in the model, Albrecht and Weber (1996) time discounting factor and uncertainty level have positive correlation.
In my model I assume a constant time discounting factor for the saved amount of consumption in the first period. In this modification the same amount of consumption in both periods are used. Hence the consumption in is represented by equation (4), where individual receives initial wealth W and spends his income on purchase of j-th asset. Equation (4) also represents the saving that individual observes at the end of the first period. This amount can be deposited under interest rate , so that in the next period risk neutral agent will receive the amount represented by equation (19) that consists of the endowment received in period represented by equation (5) and the additional income that investor receives from depositing the saved in the first period amount.
Equation (19) represents the consumption or disposable income of investor in the second period, which depends on the endowment that he receives from purchasing an asset j, which is weighted according to function represented by equation (1) and gained interest on the saved amount.
In this part of the thesis I consider risk neutral agent, as it allows analyzing more clear influence of interest rate on the equilibrium price than the equilibrium result for such modification for risk averse agents. Risk neutrality feature of agent means that agents are more concerned about the expected value of their return than the corresponding risk, hence this type of agent is indifferent between assets with same expected returns but with different risk levels. Hence his utility in each period is represented by linear function that is dependent on the consumption set in each period....