Prospect Theory and Asset Prices Presenting Barberies - Huang - Santos s paper Attila Lindner January 2009 Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 1 / 17
Presentation Outline Research question The model How does the model work? Numerical results Conclusion Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 2 / 17
Research Question To develop a GE Model that is in harmony with the following facts: low volatility of consumption growth (and i.i.d consumption growth) high mean and volatility of stock returns low and stable risk-free rate long horizon predictibility in stock returns low correlation between consumption growth and stock returns high Sharp ratio predictibility of prices using P/D reasonable risk aversion time-varying volatility of stock returns Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 3 / 17
The consumer s problem I. max E C t,s t " t=0!# ρ t C 1 γ t 1 γ + b t ρ t+1 v(x t+1, S t, z t ) s.t. : X t+1 = S t R t+1 S t R f,t C t = S t 1 R t S t R t+1 + B t 1 R f,t B t R f,t + Y t The dynamics of the state variable: R z t+1 = µ z t R t+1 + (1 µ)1 X t+1 - is the gain or loss that the agent experiences on his nancial wealth between t and t + 1 S t - the value of risky assets z t - historical benchmark factor b t - exogenous scaling factor Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 4 / 17
The consumer s problem II. Economy I (Lucas 1978): equates consumption and dividends - stocks are modeled as a claim for future consumption stream Economy II log (D t+1 /D t ) = g D + σ D ɛ t+1, where ɛ t+1 i.i.d. N(0,1) Investors have other sources of income besides dividends log (C t+1 /C t ) = g c + σ c η t+1 and log (D t+1 /D t ) = g D + σ D ɛ t+1, where η t+1 0 i.i.d. N ɛ t+1 0, 1 ϖ, ϖ 1 Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 5 / 17
Investor preferences I. b t ρ t+1 v(x t+1, S t, z t ) Capturing feelings unrelated to consumption If z t 1 (previous gain) v(x t+1, S t, z t ) = St R t+1 S t R f,t if R t+1 z t R f,t S t (z t R f,t R f,t ) + λs t (R t+1 z t R f,t ) if R t+1 < z t R f,t If z t > 1 (previous loss) St R t+1 S t R v(x t+1, S t, z t ) = f,t if X t+1 z t R f,t, λ(z t ) (S t R t+1 S t R f,t ) if X t+1 < z t R f,t where λ(z t ) = λ + k(z t 1) Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 6 / 17
Investor Preferences II. Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 7 / 17
Investor Preferences III. Historical benchmark factor z t actions taken by investors leave z t constant z t responds sluggishly z t+1 = µ z t R R t+1 + (1 µ)1 µ can be interpreted as investor memory R is the median value of returns (endogenous variable) Scaling term b t - important since consumption is increasing over time b t = b 0 Ct γ Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 8 / 17
Model Solution - Economy II. R t+1 = 1 + f (z t+1) e g d +σ D ɛ f (z t ) R f = ρ 1 e γg c γ 2 σ 2 D ɛ 1 = ρe γg d γg c +γ 2 σ 2 C 1 + f (1 ϖ2 )/2 (zt+1 ) E t e (g D γϖσ C )ɛ t+1 + f (z t ) 1 + f (zt+1 ) +b 0 ρe t bv e (g D γϖσ D )η t+1, zt f (z t ) Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 9 / 17
Model Intuition How can our model generate higher volatility in returns than in prices? Positive shock! less risk averse (z t #)! increasing prices Long horizon predictability - P/D inversely related More volatile stock prices may generate premium Price volatility driven by changes in loss aversion and by dividend shock indicating low correlation between consumption and stock returns The price of risky assets depends on volatility and not on the covariance with consumption The price volatility caused by the prospect theory part! risk free rate can be small Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 10 / 17
Price-Dividend ratio I. Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 11 / 17
Asset Prices and Returns in Economy II. Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 12 / 17
Long term predictibility (mean reversion) Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 13 / 17
Comparing results to other models Existing consumption based models do not do good jobs predicting all of these facts (especially low correlation) Di erent approach than Campbell s and Cochran s habit formation or Epstein and Zin preferences Stock returns and consumption growth are inevitably signi cantly correlated investors care about uctuations of their nancial wealth instead of covariance between risky assets and consumption (reminder: E (R i ) R f = R f Cov(m, R i )) Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 14 / 17
Sensitivity analysis Raising k has a large e ect on equity premium, since it raises average loss aversion λ increases the equity premium through increasing average loss aversion µ increases volatility of the prices and the autocorrelation of the price-dividend ratio the length of the evaluation period a ects equity premium (investors are more likely to experience losses) Using the risk free rate as reference point (opportunity cost argument) The importance of the past performance is crucial for the results Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 15 / 17
Conclusion The model can explain and predict many puzzles: equity premium, predictibility, Sharpe ratio, low correlation with consumption etc. Possible extensions: more risky assets, further understand the utility presented here contrarian vs. extrapolative expectations (it is not clear, but rather extrapolative: Durell 1999, or nothing: Shiller 1999) The "right" model should account for it too: predictibility of bonds and currencies cross-section variation in expected returns Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 16 / 17
Investor Preferences IV. These assumptions were motivated by empirical/experimental evidence: Kahneman and Tversky (1979) - prospect theory gain and loss loss aversion the gain and loss regions are linear (not the same as in standard prospect theory) using linear probabilities (not the same as in standard prospect theory) Thaler and Johnson (1990), Gertner (1993) outcomes of earlier gambles a ect risk attitudes: previous gains make agents less risk averse, previous loss make them more risk averse Assumption: investors do not integrate successive years of gains and losses (problem of mental accounting) Attila Lindner (CEU) Prospect Theory and Asset Prices January 2009 17 / 17