Salient theory and asset pricing for risk averse agents

The utility function in period 0 and 1 is represented below by set of utility equations (20). The proposed in the first chapter utility function satisfies all properties of intertemporal utility function: completeness, transitivity and continuity. Hence it is relevant to apply time discounting factor for the first period consumption in this model.

(20)

Plugging (20) in the expected utility function stated by the equation (8) and assuming unit probability of occurrence in the first period, as no uncertainty is present in this period, we obtain a new expected utility maximization problem for risk neutral agents who can deposit saved in the first period income.

(21)

The equation (21) shows the investment decision that the agent face, hence investor chooses such an amount of asset purchased that his expected utility should be maximized. The detailed derivation can be found in appendix.

is the probability of event in any states from s = 1…….S, which does not influence the equilibrium price of an asset , hence the term . Plugging this assumption back e obtain:

 (22)

Solving the first order condition for equation (22) and setting an equilibrium amount of according to BGS (2013) property for market equilibrium. We obtain equilibrium price for j-th asset which is determined by equation (23). Where states for the discounting factor for a given period. It is observed that the equilibrium price for risk neutral agents is determined by the expected dividend payoff and the covariance between dividend payoff and its weight in the portfolio discounted by the nominal interest rate on the saved from the first period amount of income. Equation (23) is provided from Bordalo, Gennaioli and Shleifer (2013) to comparing the derived equilibrium price for agents who have time preferences in depositing the saved in the first period amount with the initial model for risk neutral agents.

(23)

The logical difference between (22) and (23) is that in time of presence of discounting factor the equilibrium price is based on expected future dividends, the way agents weight the salient payoffs and then discount them by the given level of interest rate for the deposited amount. If we consider a case of non-salient payoff or rational economy, we should set from equation (1), hence all payoffs will have same weight in the portfolio and there will be no relation between dividend payoff and its weight in the portfolio according to BGS model. The equilibrium price will then be determined by the discounted expected dividend payment and represented by equation (24).

(24)

In case of non-salient payoffs, as it was mentioned in chapter 1, for the equation of weighting the payoffs (1), hence there will be no relation between dividend payments and their weights will be zero, as a result the equilibrium price for a risk neutral agent will be equal to the discounted expected dividend payment. If there is no salience present on the market risk neutral agents discount the expected value for dividend payment; intuitively this can be explained as all received payments in the future period should be adjusted to the period of occurrence, hence in the second period all cash inflows should be discounted by the given interest rate.

The same algorithm applies to the salient payoffs. For this situation agents weight the expected dividend payoffs according to the ranking of payoffs, which was proposed and discussed in equation (1). As a result the covariance is non-zero for given dividends and the equilibrium price can go up or down according to the sign of covariance. If the low payoffs are the salient ones the equilibrium price will decrease and in case of large salient payoffs the price will increase. Hence the final result is consistent with the outcome of initial Bordalo, Gennaioli and Shleifer (2013) model for risk neutral agents and the same result of mispricing is observed as for risk averse agents from chapter 1. The difference for this equilibrium price is that the sum of expected dividend payment and covariance of the payoff and its weight is marked to the time an individual receives the payment.

The effect of increase in the nominal interest rate on the equilibrium price is shown below by equation (25).

(25)

It is shown that there is negative relation between interest rate gained on saved from the first period amount and the stated in the equilibrium price if the expected dividend payment is above covariance of weight of the payoff and its value. The observed relation corresponds to the property of dependence between price and interest rate provided by Bodie, Kane, Marcus (2005), where the authors state that with the increase in interest rate the price of an asset should decrease and increase for the opposite change in interest rate. The result is consistent with the concept that Cox, Ingersoll and Ross (1985) used for their “Theory of the Term Structure if Interest Rates”. According to the authors there exists a negative relation between price of an asset and an interest rate.

However, it should be mentioned that the discussed above result will be possible if the expected level of dividend payment exceeds the covariance between weight and the payoff of the dividend. Hence if negative relation is observed on the market and negative expected dividend than the relation between interest rate and price will be positive, but this case corresponds to negative price value, which is hardly possible on the market.

To sum up the discounting factor involved in the model for salient asset pricing and risk neutral agents leads to the same results as proposed by the simple risk neutral and risk aversion model of salience, hence the mispricing will be still present on the market if agents behave irrationally and overweight large salient payoffs and underweight relatively low payoffs. The relation between interest rate and equilibrium price is found to be negative which is consistent with the empirical and theoretical models proposed by such researchers as Cox, Ingersoll and Ross (1985).

Conclusion

The modification of salient assets pricing model for different risk preferences and addition of time discounting factor are consistent with the initial model proposed by Bordalo, Gennaioli and Shleifer (2013) results as the presence of salience leads to overpricing of returns larger than average and underpricing of lower than average returns.

For the first chapter the salient assets pricing model holds for risk averse agents and proves that there exists mismatch of equilibrium price when agents are rational and weight all dividend payment equally and when some dividends have larger or lower weight due to their nature of being above or below the average dividend return. Compared to the risk neutral case of equilibrium price risk averse agents charge lower price than risk neutral agents due to existence of their preferences for risk free asset compared to risky asset, while risk neutral agents are indifferent between two outcomes. The sign of expected dividend payment influences the way agents weight its payoff and hence the equilibrium price leading to underpricing of low expected return assets and overpricing of high expected returns.

In the second part of first chapter for the risk lover case the main finding of salience theory still holds, as the mismatch between prices exists due to presence of non-rational weighting of the assets payoffs. Another important finding concerning model for risk loving agents and risk averse states that risk loving agents will charge higher price in case of positive returns, hence the overpricing feature is even higher for risk lovers as for risk averse agents. And the underpricing is stronger for risk averse agents due to the sign of risk aversion coefficient and their nature of protecting themselves from risk. This result is supported by Bordalo, Gennaiolli and Shleifer (2010) analysis of risk preferences behavior of agents, which states that risk loving agents pay attention to highest lottery payoffs while risk averse agents draw attention to lowest lottery payoffs.

The second chapter results suggest that there is positive relation between risk aversion coefficient and equilibrium price in case of positive expected dividend payments and negative relation in case of expected dividends being below zero, hence the covariance of such dividend with its weight is expected to be also below zero and stating that the lowest dividend payments are the most salient ones. Intuitively the more risk verse is an agent the higher risk premium he requires for investment decision, hence the more he will prefer risk free asset to a risky one. In case of non-salient assets the discussed effect of variation in risk aversion leads to smaller magnitude of change in equilibrium price. As a result salience presence lead to larger mismatch between fundament price and equilibrium price in the market.

The analyzed wealth effect implies that there is always a negative relation between the equilibrium price and the initial level of wealth for the risk averse agent for both salient and non-salient payoffs. This could be intuitively explained as agents are less willing to scarify large amount of wealth for dividends and hence the demand for dividends will fall leading to lower equilibrium price. The more agents receive in the first period the more they are interested in investing in risk free assets according to their risk preferences, as due to Rabin and Thaler (2001) the agents are more loss averse than risk averse, hence observing larger initial wealth they can loose larger amount in case of investing in risky asset. This leads to the negative relation between wealth and equilibrium price in the model.

The time discounting factor for the risk neutral model of salient asset pricing is consistent with the overall conclusion about salience theory, that the price of dividends could be overpriced in case of large dividends payments and underpriced in case of low dividend payments. For this modification agents discount the expected dividend payment and the relation between payment and its weight in the portfolio, that is logical as the equilibrium price should correspond to the ability of agents to deposit saved in the first period amount and adjust all cash inflows to particular period of time. Hence agents discount the equilibrium price to adjust it to the next period of time and get the so called fair value of an asset. It case of non-salient payoffs, the equilibrium price of an asset will be the discounted expected dividend payment, which is consistent with stated price of a bond formula in Bodie, Kane, Marcus (2005).

The overall result of the solved model is useful for determining the reasons of growth-value puzzles, can explain why better performing stocks are overpriced and the opposite situation applies for the low performance stocks. It also states that the risk preferences variation leads to changes in the magnitude of mispricing, hence for risk loving agents the mispricing is stronger than for risk averse or risk neutral agents. Risk averse agents showed the lower value of mispricing due to their willingness to invest in more certain assets than in uncertain prospects.

The proposed in this work model experiences several limitations and gives rise for further development. Firstly, the assumption linear probability weights in the model is under criticism as Allais (1953) questioned whether people actually choose using linear probability weights, and Tversky and Kahneman (1981) showed that people's choices can vary depending on the wording (or "framing") of a problem, rather than its objective features, hence proposed in the first chapter model could be discussed under different assumptions of probability features. Secondly, as Rabin, M. (2000) argues that expected utility theory is misleading in explaining the variation in risk aversion coefficient due to scale of the payoffs, as the variation can be some irregular feature of the model formed from the specific feature of the data analyzed rather than typical theoretically stated assumption of the model. And lastly, the last chapter derived a private case for depositing additional savings for the last period for risk averse agents for model simplification. For the further development it is interesting to analyze the effect of discounting factor for all types of risk preferences and show for which type of agents the mispricing would perform the highest results and what the effect of interest rate would be on the equilibrium price. Moreover, the time discounting factor in my model is assumed to be constant, while the financial market experiences such anomalies as time varying discounting factor connected with the outcome size or level of uncertainty in the model.

List of References

1. Albrecht, Martin and Martin Weber. 1996. The resolution of uncertainty: An experimental study. Journal of Institutional and Theoretical Economics, 152:4, pp. 593-607.

2. Allais, Maurice. 1953 Le comportement de l'homme rationnel devant le risque: critique des postulats et axiomes de l'Ecole Americaine.”Econometrica, 21: 503-546.

3. Arrow. 1971. “Essays in the Theory of Risk-Bearing”. Chicago, IL: Markham Publishing Company.

4. Barberis, Greenwood, Jin and Shleifer. 2015. X-CAPM: An extrapolative capital asset pricing model Journal of Financial Economics Volume 115, Issue 1, January 2015, Pages 1-24

5. Barberis, Nicholas, Ming Huang and Tano Santos. Prospect Theory And Asset Prices," Quarterly Journal of Economics, 2001, v116(1,Feb), 1-53

6. Barberis, Nicholas, Ming Huang, and Tano Santos. 2001. Prospect Theory and Asset Prices. Quarterly Journal of Economics 116 (1): 1-53.

7. Benartzi, Shlomo, and Richard H. Thaler. 2000. Myopic Loss Aversion and the Equity Pre- mium Puzzle. New York: Cambridge University Press.

8. Benzion, Uri, Amnon Rapoport, and Joseph Yagil. 1989. “Discount rates inferred from decisions: An experimental study. Management Science, 35, pp. 270-84.

9. Bodie, Kane, Marcus. 2005. “Investments” Sixth Edition. McGraw-Hill Part 4, Chapter 14, pp 447-519

10. Bordalo, Pedro, Nicola Gennaioli, and Andrei Shleifer .2010. Salience Theory of Choice Under Risk. National Bureau of Economic Research 1050 Massachusetts Avenue?Cambridge, MA 02138?

11. Bordalo, Pedro, Nicola Gennaioli, and Andrei Shleifer. 2012a. Salience Theory of Choice under Risk. Quarterly Journal of Economics 127 (3):1243-85.

12. Bordalo, Pedro, Nicola Gennaioli, and Andrei Shleifer. 2012b. Salience and Consumer Choice. National Bureau of Economic Research Working Paper 17947.

13. Bordalo, Pedro, Nicola Gennaioli, and Andrei Shleifer .2013. Salience and Asset Prices. American Economic Review: Papers&Proceedings 2013, 103(3): 623-628

14. Campbell, John Y., and Robert Shiller. 1988. The Dividend-Price Ratio and Expectations of Future Dividends and Discount Factors, Review of Financial Studies, 1 (3): 195-228.

15. Cox, J.C., J.E. Ingersoll and S.A. Ross (1985). A Theory of the Term Structure of Interest Rates. Econometrica 53: 385-407. doi: 10.2307/1911242.

16. De Long, Shleifer, Summers and Waldmann. 1990. Noise Trader Risk in Financial Markets. Journal of Political Economy Volume 98, Number 4, August 1990

17. Fama, Eugene F., and Kenneth R. French. 1992. “The Cross-Section of Expected Stock Returns” Journal of Finance 47 (2):427-65.

18. Fisher. 1930. The Theory of Interest, As Determined by Impatience to Spend Income and Opportunitity to Invest It. New York: Macmillan, 1930

19. Frederick, George Loewenstein and Ted O'Donoghue. 2002. Time Discounting and Time Preference: A Critical Review Journal of Economic Literature Vol. 40, No.2 (Jun., 2002), pp. 351-401

20. Fullenkamp, Tenorio and Battalio. 2003. Assessing individual risk attitudes using field data from lottery games. Review of Economics and statistics, 2003 - MIT Press

21. Kahneman and Lovallo (1993), Timid Choices and Bold Forecasts: A Cognitive Perspective on Risk Taking, Management Science, 39, 17 31.

22. Kahneman, Daniel, and Amos Tversky. 1979. Prospect Theory: An Analysis of Decision under Risk. Econometrica 47 (2):263-91.

23. Lakonishok, Josef, Andrei Shleifer, and Robert W. Vishny. 1994. Contrarian Investment, Extrapolation, and Risk. Journal of Finance 49 (5):1541-78.

24. Marjorie K. Shelley. 1993. Outcome Signs, Question Frames and Discount Rates. Department of Accountancy, University of Illinois, Urbana-Champaign, Illinois 61820

25. Mehra, Rajnish, and Edward C. Prescott. 1985. The Equity Premium: A Puzzle.” Journal of Monetary Economics 15 (2): 145-61.

26. Pratt, J.W. and R.J. Zeckhauser. 1987: Proper Risk Aversion, Econometrica, 55, 143-154.

27. Pratt. J.W. 1964. Risk Aversion in the Small and in the Large. Econometrica Vol. 32, No. 1/2 (Jan. - Apr., 1964), pp. 122-136

28. Rabin, Mathew, Risk aversion and expected-utility theory: A calibration theorem. Econometrica, 68 (2000): 1281-1292.

29. Rabin M. and. Thaler. R.H. 2001. Anomalies: Risk Aversion The Journal of Economic Perspectives Vol. 15, No.1 (Winter, 2001), pp. 219-232

30. Read Daniel, Loewenstein George, Rabin Matthew, Keren Gideon, Laibson David.1999. Choice Bracketing. Springer Netherlands, pp 171-202

31. Shleifer Andrei and Vishny Robert W. 1990. Equilibrium Short Horizons of Investors and Firms. The American Economic Review Vol. 80, No.2, Papers and Proceedings of the Hundred and Second Annual Meeting of the American Economic Association (May, 1990), pp. 148-153

32. Taylor, Shelley, and Suzanne Thompson. 1982. Stalking the elusive vividness effect. Psychological Review, 89(2): 155-181.?

33. Thaler .1981. Some Empirical Evidence on Dynamic Inconsistency. Quasi rational economics, 1991, pp 127-132

34. Thaler. 1999. Mental accounting matters. Journal of Behavioral Decision Making Close: block: publication Block 12.3 (Sep 1999): 183.

35. Tversky, Amos and Daniel Kahneman. 1981. The framing of decisions and the psychology of choice. Science, 211(4481): 453-458.

36. Tversky (1992) Advances in Prospect Theory: Cumulative representation of uncertainty. Journal of Risk and Uncertainty, 5 (1992): 297-323.

Appendix 1

Chapter 1. Maximization of risk averse agent's expected utility. Equation (8).

(6)

(7)

(8)

According to Bordalo, Gennaioli and Shleifer the equilibrium buying decision for investors is determined by , which comes from maximizing the expected utility, represented above by equation (8) and market equilibrium occurs at =0 for all j. hence plugging in (25) =0 we obtain equation (26).

(11)

Appendix 2

Analyzing the derivative of equilibrium price with respect to risk aversion coefficient - equation (13).

(13)

Properties:

If , than

Hence the whole derivative is positive and the same movements in risk aversion and equilibrium price will be obtained.

If , than

Hence the whole derivative is below zero in case of negative returns being present on the market.

Appendix 3

Comparison of risk averse and risk lover equilibrium prices:

For risk loving agents:

For risk averse agents:

The equation above can be rewritten as

vs

>, for any

It follows that the price for risk loving gents is larger than the price for risk averse agents.

Appendix 4

Discount Factor model of salient payoffs for risk neutral agents. Maximization of equation (21).

Setting as BGS state for the equilibrium on the market, same as in the first chapter of the thesis.

is the probability of event in any states from s=1…….S, which does not influence the equilibrium price of an asset , hence the term . Plugging this assumption back e obtain:

Let

(22)

Размещено на Allbest.ru