Behavioral Finance and Asset Pricing

Behavioral Finance and Asset Pricing Behavioral Finance and Asset Pricing /49

Introduction We present models of asset pricing where investors preferences are subject to psychological biases or where investors make systematic errors in judging the probability distribution of asset returns. A model that incorporates some form of irrationality attempts to provide a positive or descriptive theory of individual behavior (behavioral nance). We rst consider Barberis, Huang, and Santos (200) model of an endowment economy where investors decisions exhibit prospect theory. Second, we examine the model of Kogan, Ross, Wang, and Wester eld (2006) where some investors su er systematic optimism or pessimism. Behavioral Finance and Asset Pricing 2/49

Prospect Theory Prospect Theory (Kahneman and Amos Tversky (979)) speci es investor utility that is a function of recent changes in, rather than simply the current level of, nancial wealth. An example is loss aversion which characterizes investor utility that is more sensitive to recent losses than recent gains in nancial wealth. A related bias is the house money e ect which characterizes utility where losses following previous losses create more disutility than losses following previous gains: After a run-up in asset prices, the investor is less risk averse because subsequent losses would be cushioned by the previous gains. Behavioral Finance and Asset Pricing 3/49

Implications of Prospect Theory As shown by the Barberis, Huang, and Santos model, loss aversion together with the house money e ect have implications for the dynamics of asset prices. After a substantial rise in asset prices, lower investor risk aversion can drive prices even higher, making asset price volatility exceed that of fundamentals (dividends). These biases also generate predictability in asset returns since a substantial recent fall (rise) in asset prices increases (decreases) risk aversion and expected asset returns. These biases can also imply a high equity risk premium because the excess volatility in equity prices leads loss-averse investors to demand a relatively high average rate of return on equities. Behavioral Finance and Asset Pricing 4/49

Barberis, Huang, Santos Model Assumptions Technology: There is a discrete-time endowment economy where the risky asset portfolio pays a date t perishable dividend of D t. Date t aggregate consumption, C t, equals this dividend, D t, plus perishable non nancial income, Y t. C t and D t, follow the joint lognormal process ln C t+ =C t = gc + C t+ () ln (D t+ =D t ) = g D + D " t+ where t+ and " t+ are serially uncorrelated and distributed t 0 ~N ; (2) 0 " t Behavioral Finance and Asset Pricing 5/49

Assumptions (continued) Let the risky asset return from date t to date t + be R t+ (P t+ + D t+ ) =P t, and let the zero-net supply risk-free asset return from t to t + be R f ;t. Preferences: Representative, in nitely lived individuals maximize " X E 0 t C t # + b t t+ v (X t+ ; w t ; z t ) (3) t=0 where C t is the individual s consumption. w t is the number of shares of the risky asset held by the individual at date t. Behavioral Finance and Asset Pricing 6/49

Assumptions (continued) X t+ is the total excess return earned on the risky asset from t to t + and is de ned as X t+ w t (R t+ R f ;t ) (4) z t < (>) measures prior gains (losses) on the risky asset: z t = ( ) + z t R R t (5) where 0 and R is a parameter, approximately equal to the average risky-asset return. The greater is, the longer is the investor s memory in measuring gains from the risky asset. Behavioral Finance and Asset Pricing 7/49

Assumptions (continued) v () models prospect theory s e ect of risky-asset gains/losses. If z t = (no prior gains/losses), v () displays pure loss aversion: Xt+ if X v (X t+ ; w t ; ) = t+ 0 X t+ if X t+ < 0 (6) where >. If z t 6=, v () re ects the house money e ect. For prior gains (z t ), it equals = Xt+ if R t+ z t R f ;t X t+ + ( ) w t (R t+ z t R f ;t ) if R t+ < z t R f ;t v (X t+ ; w t ; z t ) (7) Behavioral Finance and Asset Pricing 8/49

Assumptions (continued) For prior losses (z t > ), it equals v (X t+ ; w t ; z t ) = Xt+ if X t+ 0 (z t ) X t+ if X t+ < 0 (8) where (z t ) = + k (z t ), k > 0. Losses that follow previous losses are penalized at the rate of (z t ), which exceeds. The prospect theory term in the utility function is scaled to make the risky asset price-dividend ratio and the risky asset risk premium stationary with increases in aggregate wealth: where b 0 > 0. b t = b 0 C t (9) Behavioral Finance and Asset Pricing 9/49

Solving the Model The state variables for the individual s consumption-portfolio choice problem are wealth, W t, and z t. We assume f t P t =D t = f t (z t ) and then show that an equilibrium exists in which this is true. Hence, the return on the risky asset can be written R t+ = P t+ + D t+ P t = + f (z t+) e g D + D " t+ f (z t ) = + f (z t+) D t+ (0) f (z t ) D t Let R f ;t = R f, a constant, which will be veri ed by the solution to the agent s rst-order conditions. Making this assumption simpli es the form of v (). Behavioral Finance and Asset Pricing 0/49

Solution (continued) Note from (7) and (8) that v (X t+ ; w t ; z t ) can be written v (X t+ ; w t ; z t ) = w t bv (R t+ ; z t ), where for z t < = Rt+ R f if R t+ z t R f R t+ R f + ( ) (R t+ z t R f ) if R t+ < z t R f bv (R t+ ; z t ) () and for z t > bv (R t+ ; z t ) = Rt+ R f if R t+ R f (z t ) (R t+ R f ) if R t+ < R f (2) Behavioral Finance and Asset Pricing /49

Solution (continued) The individual s maximization problem is then " X max fc t ;w t g E 0 t=0 subject to the budget constraint t C t + b 0 t+ C t w t bv (R t+ ; z t ) # (3) W t+ = (W t + Y t C t ) R f + w t (R t+ R f ) (4) and the dynamics for z t given in (5). Behavioral Finance and Asset Pricing 2/49

Derived Utility of Wealth De ne t J (W t ; z t ) as the derived utility-of-wealth function. Then the Bellman equation for this problem is Ct J (W t ; z t ) = max (5) fc t ;w t g h i +E t b 0 C t w t bv (R t+ ; z t ) + J (W t+ ; z t+ ) Behavioral Finance and Asset Pricing 3/49

First Order Conditions Di erentiating with respect to C t and w t : 0 = C t R f E t [J W (W t+ ; z t+ )] (6) h i 0 = E t b 0 C t bv (R t+ ; z t ) + J W (W t+ ; z t+ ) (R t+ R f ) = b 0 C t E t [bv (R t+ ; z t )] + E t [J W (W t+ ; z t+ ) R t+ ] R f E t [J W (W t+ ; z t+ )] (7) Behavioral Finance and Asset Pricing 4/49

Solution (continued) It is straightforward to show that (6) and (7) imply the standard envelope condition C t = J W (W t ; z t ) (8) Substituting this into (6), one obtains the Euler equation " Ct+ # = R f E t (9) Using () to compute the expectation in (9), we can solve for the risk-free interest rate: C t R f = e ( )g C 2 ( )2 2 C = (20) Behavioral Finance and Asset Pricing 5/49

Solution (continued) Using (8) and (9) in (7) implies h i 0 = b 0 C t E t [bv (R t+ ; z t )] + E t C t+ R t+ h i = b 0 C t E t [bv (R t+ ; z t )] + E t C t+ R t+ R f E t h C t+ i C t = (2) or " # C t Ct+ = b 0 E t [bv (R t+; z t )] + E t R t+ C t C t (22) In equilibrium, (9) and (22) hold with individual consumption, C t, replacing aggregate per-capita consumption, C t. Behavioral Finance and Asset Pricing 6/49

Solution (continued) Using () and (0), (22) is simpli ed to: or = b 0 E t [bv (R t+ ; z t )] (23) + f (zt+ ) +E t e g D + D " t+ e g C + C t+ f (z t ) = + f (zt+ ) b 0 E t bv e g D + D " t+ ; z t f (z t ) +e g D ( )g C + 2 ( )2 2 C ( 2 ) + f (zt+ ) E t e ( D ( ) C )" t+ f (z t ) (24) Behavioral Finance and Asset Pricing 7/49

Solution (continued) The price-dividend ratio, P t =D t = f t (z t ), can be computed numerically from (24). R However, because z t+ = + and z t R t+ R t+ = +f (z t+) f (z t ) e g D + D " t+, z t+ depends upon z t, f (z t ), f (z t+ ), and " t+ : Rf (z t ) e g D D " t+ z t+ = + z t + f (z t+ ) (25) Therefore, (24) and (25) need to be solved jointly and can be done by an iterative numerical technique for nding the function f (). Behavioral Finance and Asset Pricing 8/49

Numerical Soution for Price/Dividend Ratio Start by guessing an initial function, f (0), and use it to solve for z t+ in (25) for given z t and " t+. Then, a new candidate solution, f (), is obtained using the following recursion that is based on (24): f (i+) (z t ) = e g D ( )g C + 2 ( )2 2 C ( 2 ) i i E t hh + f (i) (z t+ ) e ( D ( ) C )" t+ +f (i) (z t ) b 0 E t "bv!# + f (i) (z t+ ) f (i) e g D + D " t+ ; z t (z t ) where the expectations are computed using a Monte Carlo simulation of the " t+. Given f (), z t+ is solved again from (25) and the procedure is repeated until f (i) converges. Behavioral Finance and Asset Pricing 9/49

Model Results For reasonable parameter, P t =D t = f t (z t ) decreases in z t : if there are prior risky asset gains (z t is low), then investors are less risk averse and bid up the risky asset price. Using the estimated f (), the unconditional distribution of stock returns is simulated by randomly generating " t s. This shows that since dividends and consumption follow separate processes, and stock prices have volatility exceeding that of dividends (fundamentals), stock volatility can be made substantially higher than consumption volatility. Behavioral Finance and Asset Pricing 20/49

Model Results (continued) Moreover, the e ect of loss aversion generates a signi cant equity risk premium for reasonable values of. Because investors care about stock volatility, per se, a large equity premium can exist despite low stock-consumption correlation. Consistent with empirical research nding negative correlations in stock returns at long horizons, the model generates predictability in stock returns: returns tend to be higher following crashes (when z t is high) and smaller following expansions (when z t is low). Behavioral Finance and Asset Pricing 2/49

The Impact of Irrational Traders on Asset Prices The Kogan, Ross, Wang, and Wester eld (2006) model assumes some investors are fully rational but others are irrational because they su er from systematic optimism or pessimism. The model shows that irrational investors may not necessarily lose wealth to rational investors and be driven out of the asset market. Even when irrational investors do not survive in the long run, their trading can signi cantly impact equilibrium asset prices for substantial periods. Behavioral Finance and Asset Pricing 22/49

Kogan, Ross, Wang, Wester eld Model Assumptions A simple endowment economy has two types of representative agents: rational agents and agents that are irrationally optimistic or pessimistic regarding risky-asset returns. Both maximize utility of consumption at a single, future date. Technology: The risky asset is a claim on a single, risky date T > 0 dividend payment, D T. D T is the date T realization of dd t =D t = dt + dz (27) where and are constants, > 0, and D 0 =. Aggregate consumption at date T is C T = D T. All agents can buy or sell (issue) a zero-coupon bond in zero net supply that makes a default-free payment of at date T. Behavioral Finance and Asset Pricing 23/49

Assumptions (continued) Preferences and Beliefs: Rational and irrational agents each have date 0 endowment equal to one-half of the risky asset and have constant relative risk aversion. For example, the rational agents maximize " # C r ;T E 0 (28) where < and C r ;T is rational traders date T consumption. While rational agents believe (27), irrational agents perceive dd t =D t = + 2 dt + dbz (29) where they believe dbz is a Brownian motion, whereas in reality, dbz = dz dt. Note if the constant is positive (negative), irrational traders are optimistic (pessimistic). Behavioral Finance and Asset Pricing 24/49

Irrational Agent Beliefs Hence, rather than the probability measure P that is generated by dz, irrational traders believe that the probability measure is generated by dbz, which we refer to as the probability measure b P. Therefore, an irrational individual s expected utility is " # C n;t be 0 (30) where C n;t is the date T consumption of the irrational trader. Behavioral Finance and Asset Pricing 25/49

Solution Technique The irrational agent s utility can be reinterpreted as the state-dependent utility of a rational individual. Girsanov s theorem implies d b P T = ( T = 0 ) dp T where if 0 =, then Z T T = exp dz 0 = e 2 2 2 T +(z T z 0 ) Z T () 2 ds 2 0 (3) Since and are constants, t is lognormal d= = dz. Behavioral Finance and Asset Pricing 26/49

Utility of Irrational Agents Thus, the irrational agents s expected utility can be written as " # " C n;t C # n;t be 0 = E 0 T (32) # = E 0 " e 2 2 2 T +(z T z 0 ) C n;t (32) shows that the objective function of the irrational trader is observationally equivalent to that of a rational trader whose utility depends on z T, which is the same state (Brownian motion uncertainty) determining the risky asset s dividend. Behavioral Finance and Asset Pricing 27/49

Martingale Approach to Consumption and Portfolio Choice Given market completeness, the martingale approach where lifetime utility contains only a terminal bequest can be applied. The two types of agents rst order conditions are C r ;T = r M T (33) T C n;t = n M T (34) where r and n are the Lagrange multipliers for the rational and irrational agents, respectively. Substituting out for M T, we can write C r ;T = ( T ) C n;t (35) where we de ne r = n. Behavioral Finance and Asset Pricing 28/49

Market Equilibrium Market clearing at the terminal date implies C r ;T + C n;t = D T (36) Equations (35) and (36) allow us to write: C r ;T = + ( T ) D T (37) Substituting (37) into (35), we also obtain C n;t = ( T ) D + ( T ) T (38) Behavioral Finance and Asset Pricing 29/49

Solution The parameter = r = n is determined by the individuals initial endowments of wealth, equal to E 0 [C i;t M T =M 0 ], i = r; n. Note that the date t price of the zero coupon bond that pays at date T > t is P (t; T ) = E t [M T =M t ] (39) For analytical convenience, consider de ating all assets prices, including the individuals initial wealths, by this zero-coupon bond price. De ne W r ;0 and W n;0 as the initial wealths, de ated by this zero-coupon bond price, of the rational and irrational individuals, respectively. Behavioral Finance and Asset Pricing 30/49

De ated Initial Wealths The de ated wealth of the rational agent is W r ;0 = E 0 [C r ;T M T =M 0 ] E 0 [M T =M 0 ] h E 0 C r ;T C = = r ;T = r E 0 h C r ;T = r = E 0 [C r ;T M T ] E 0 [M T ] i i E 0 hc r ;T i = h i E 0 C r ;T E 0 h + ( T ) i D T E 0 h + ( T ) i D T (40) where in the second line of (40), (33) is used to substitute for M T, and in the third line (37) is used to substitute for C r ;T. Behavioral Finance and Asset Pricing 3/49

Solution for Lagrange Multiplier A similar derivation that uses (34) and (38) leads to i E 0 ( T ) h + ( T ) D T W n;0 = h i E 0 + ( T ) (4) D T Since agents begin with equal 2 shares of the endowment, W r ;0 = W n;0. Equating the right-hand sides of (40) and (4) and noting that T satis es (3) and D T =D t = e [ 2 2 ](T t)+(z T z t ) is also lognormally distributed, it can be shown that = e 2 T Behavioral Finance and Asset Pricing 32/49 (42) (43)

Price of the Risky Asset Given, M T =M t is a constant times i h + ( T ) D T, which allows us to solve for the equilibrium price of the risky asset. De ne S t as the date t < T price of the risky asset de ated by the price of the zero-coupon bond, and de ne " T ;t T = t e 2 T 2 2 2 (T t)+(z T z t ). Then S t = E t [D T M T =M t ] E t [M T =M t ] = E t " + " E t " + " T ;t T ;t D T # D T # (44) Behavioral Finance and Asset Pricing 33/49

Analysis of the Results Though the rational and irrational agents portfolio policies do not have a closed form solution, it can be shown that agents demand for the risky asset,!, satis es j!j + jj (2 ) = ( ). For the case of all rational agents, = 0, then " T ;t = t = and from (44) the de ated stock price, S r ;t, is S r ;t = E t D h T i = D t e [ 2 ](T t)+ 2 (T t) (45) E t D T = e [ ( )2 ]T +[( ) Itô s lemma shows that (45) implies: 2 ] 2 t+(z t z 0 ) ds r ;t =S r ;t = ( ) 2 dt + dz (46) Behavioral Finance and Asset Pricing 34/49

Results (continued) Similarly, when all agents are irrational, S n;t satis es S n;t = e [ ( )2 ]T +[( ) 2 ] 2 t+(z t z 0 ) 2(T t) = S r ;t e (47) and its rate of return follows the process ds n;t =S n;t = ( ) 2 dt + dz (48) Note that the e ect of is similar to, so that if is positive, the higher expected dividend growth acts like lower risk aversion. The greater demand raises the de ated stock price and lowers its equilibrium expected rate of return. Behavioral Finance and Asset Pricing 35/49

Results (continued) Note that (46) and (48) indicate that when there is only one type of agent, the volatility of the risky asset s de ated return equals. In contrast, when both types of agents are in the economy, applying Itô s lemma to (44) it can be shown that the risky asset s volatility, S;t, satis es S;t ( + jj) (49) The conclusion is that a diversity of beliefs has the e ect of raising the equilibrium volatility of the risky asset. Behavioral Finance and Asset Pricing 36/49

Risky Asset Price with Logarithmic Utiliy When utility is logarithmic so that = 0, (44) simpli es to S t = + E t [ T ] E t ( + T ) D (50) T = D t e [ 2 ](T t) + t + t e 2 (T t) = e [ 2 2 ]T 2 2 (T t)+(z t z 0 ) + t + t e 2 (T t) Behavioral Finance and Asset Pricing 37/49

Rational Agents Share of Wealth De ne t = t W r ;t W r ;t + W n;t = W r ;t S t (5) as the proportion of total wealth owned by the rational individuals. Using (40) and (44), when = 0 it equals " # E t + " T ;t D T E t " + " T ;t D T # = + E t [ T ] = + t (52) Behavioral Finance and Asset Pricing 38/49

Mean and Volatility of Risky Asset with Log Utility Viewing S t as a function of D t and t as in the second line of (50), Itô s lemma can be applied to derive " # S;t = + + e 2 (T t) t t (53) and S;t = 2 S;t ( t ) S;t (54) where we have used t = = ( + t ) to substitute out for t. Note that when t = or 0, (53) and (54) are consistent with (46) and (48) for the case of = 0. Behavioral Finance and Asset Pricing 39/49

Friedman Conjecture The model is used to study how C n;t =C r ;T is distributed as T becomes large. Milton Friedman (953) conjectured that irrational traders cannot survive in a competitive market: the relative extinction of an irrational agent would occur if lim T! C n;t C r ;T = 0 a.s. (55) which means that for arbitrarily small the probability of lim C n;t C r ;T > equals zero. T! An agent is said to survive relatively in the long run if relative extinction does not occur. Behavioral Finance and Asset Pricing 40/49

Survival/Extinction under Log Utility For log utility, irrational agents always su er relative extinction. To see this, rearrange (35): C n;t C r ;T = ( T ) (56) and for = 0, (43) implies that =. Hence, C n;t C r ;T = T (57) = e 2 2 2 T +(z T z 0 ) Behavioral Finance and Asset Pricing 4/49

Stong Law of Large Numbers Based on the strong law of large numbers for Brownian motions, it can be shown that for any value of b lim T! eat +b(z T z 0 ) = 0 a < 0 a > 0 (58) where convergence occurs almost surely. Since 2 2 2 < 0 in (57), equation (55) is proved. The intuition for relative extinction is linked to the specialness of log utility. The logarithmic rational agent maximizes at each date t: E t [ln C r ;T ] = E t [ln W r ;T ] (59) Behavioral Finance and Asset Pricing 42/49

Growth-Optimum Portfolio (59) is equivalent to maximizing the expected continuously compounded return: E t T t ln (W r ;T =W r ;t ) = T t [E t [ln (W r ;T )] ln (W r ;t )] since W r ;t is known at date t and T t > 0. Thus, this portfolio policy maximizes E t [d ln W r ;t ] and is referred to as the growth-optimum portfolio. Note that the rational and irrational agents wealths satisfy (60) dw r ;t =W r ;t = r ;t dt + r ;t dz (6) dw n;t =W n;t = n;t dt + n;t dz (62) where, in general, r ;t, n;t, r ;t, and n;t, are time varying. Behavioral Finance and Asset Pricing 43/49

Growth in Relative Wealths Applying Itô s lemma, it is straightforward to show Wn;t d ln = n;t 2 2 n;t r ;t 2 2 r ;t dt W r ;t + ( n;t r ;t ) dz = E t [d ln W n;t ] E t [d ln W r ;t ] + ( n;t r ;t ) dz Since the irrational agents choose a portfolio policy that deviates from the growth-optimum portfolio, we know E t [d ln W n;t ] E t [d ln W r ;t ] < 0, and thus E t [d ln (W n;t =W r ;t )] < 0, making d ln (W n;t =W r ;t ) a process that is expected to steadily decline as t!, verifying Friedman s conjecture. (63) Behavioral Finance and Asset Pricing 44/49

Survival for General CRRA Utility The presence of irrational agents can impact asset prices for substantial periods of time prior to becoming "extinct." Moreover, if < 0 Friedman s conjecture may not always hold. Computing (56) for the general case of = e 2T : C n;t C r ;T = ( T ) (64) = e [+ 2 2 ] 2 T + (z T z 0 ) The limiting behavior of C n;t =C r ;T depends on the sign of + 2 2 or + 2. Behavioral Finance and Asset Pricing 45/49

Survival/Extinction for General CRRA Utility If < 0, the strong law of large numbers implies 8 C < 0 < 0 rational trader survives n;t lim = 0 < < 2 irrational trader survives T! C r ;T : 0 2 < rational trader survives (65) If the irrational agent is pessimistic ( < 0) or strongly optimistic ( > 2), he becomes relatively extinct. However, when the irrational agent is moderately optimistic (0 < < 2), it is the rational agent who becomes relatively extinct! Behavioral Finance and Asset Pricing 46/49

Intuition for General Result The intuition is that when < 0, rational agents demand for the risky asset is less than that of a log utility agent, so that their wealths grow more slowly. When the irrational agent is moderately optimistic (0 < < 2), her portfolio demand is relatively closer to the growth-optimal portfolio. Behavioral Finance and Asset Pricing 47/49

Extensions If agents were assumed to gain utility from interim consumption, this would reduce the growth of their wealth and a ect their relative survivability. Also, systematic di erences between rational and irrational agents risk aversions could in uence the model s conclusions. In addition, one might expect that irrational agents might learn over time of their mistakes. Lastly, the model considers only one form of irrationality: systematic optimism or pessimism. Behavioral Finance and Asset Pricing 48/49

Summary This note considered two equilibrium models that incorporate psychological biases or irrationality. While considered behavioral nance models, they can be solved using standard techniques. Currently, there is little consensus among nancial economists regarding the importance of incorporating aspects of behavioral nance into asset pricing theories. Behavioral Finance and Asset Pricing 49/49