Behavioral Finance and
Asset Pricing
What effect does psychological bias
(irrationality) have on asset demands
and asset prices?
1. We want to employ an intertemporal choice model to
evaluate the effect of psychological bias on consumption and
portfolio choices
Barberis, Huang and Santos (2001): Under prospect theory we can identify
two forms of psychological bias - • loss aversion – investors are more sensitive to recent losses than they are
to recent gains in wealth
• house money effect – the sequencing of losses and gains has an effect on
investor behavior, so that after a recent run-up in asset prices (gains)
investors become less risk averse (gains “cushion” losses)
Premise of the BHS model:
• investors derive direct utility from consumption and changes in the value
of financial wealth.
• investors care about fluctuations in financial wealth whether or not they
are correlated with consumption.
• investors have time-varying risk aversion due to loss aversion from
financial wealth fluctuations
Implications of the BHS model:
• asset prices exhibit greater volatility than they would based solely on
changes in fundamentals (earnings).
• asset prices and returns become more predictable.
• high equity risk premium (“excess volatility” in asset prices leads risk
averse investors to require a higher rate of return on stocks).
BHS model economy 1
(Lucas 1978 model where consumption is equated to dividends,
so that stocks are a claim on future consumption)
Technology
• A portfolio of risky assets has return (Rt+1) and pays a stream of dividends
(Dt). In equilibrium aggregate consumption (Ct) equals dividends plus
nonfinancial income from labor (Yt). Both dividends and nonfinancial
income are perishable. The risk-free asset has rate Rf,t .
• Aggregate consumption and dividends follow a joint lognormal process
where the error terms ~ i.i.d. N(0,1) as
Preferences
• Infinitely lived individuals maximize their lifetime utility
where γ < 1 is the risk aversion coefficient, ρ is a time discount factor, St is
the value of risky asset at date t, Xt+1 is the total excess return from
holding the risky asset (relative to the return on the risk-free asset),
bt is a “scaling” parameter. zt is a state variable. It is the individual’s prior
gains (losses) as a fraction of St. zt = Zt/St where Zt is the historical
“benchmark” level of stock price that is identified by the investor. zt < 1
indicates a prior gain occurred, zt > 1 indicates a prior loss occurred.
zt follows a process where 0≤η≤1 and the average R makes zt = 1 in steady
state. With larger η the investor’s memory is longer.
• The function v ( ) characterizes the prospect theory effect of risky asset
gains on utility.
• If zt = 1 (no prior gains or losses) we get pure loss aversion where λ > 1 and
losses have a bigger effect on utility.
• When zt ≠ 1 the v( ) function reflects the house money effect.
• If zt ≤ 1 (prior gains) the v( ) function becomes,
Interpretation of (15.7): When a return exceeds the cushion built in by prior
gains (Rt+1 ≥ ztRf,t) the effect on utility is 1:1. When the gain is less than the
amount of prior gains (Rt+1<ztRf,t) the effect on disutility is greater than 1:1.
• If zt > 1 (prior losses)) the v( ) function becomes
where λ(zt) = λ + k(zt -1) and k > 0. Losses that follow previous losses are
penalized at the rate λ(zt), which exceeds λ and grows larger as losses
become larger (as zt >1).
• The prospect theory term, v( ), is scaled by bt to make the risky asset
price/dividend ratio and the risky asset risk premium stationary as wealth
increases over time.
Prospect theory effect of risky asset
gains and losses on utility
2. We want to solve the model to show that an equilibrium exists
•
•
•
•
The state variables of the consumption-portfolio choice model are wealth (Wt)
and zt.
Assume that the ratio of asset price/dividend is a decreasing function of zt
where ft = Pt/Dt = f(zt). We want to show that an equilibrium exists where that
is true.
Assuming that Rf is constant, we can write the return on the risky asset as
From (15.7) and (15.8) we can verify that v( ) is proportional to St. Thus,
v(Xt+1, St, zt) can be rewritten as Stv(Rt+1, zt). If zt < 1 (prior gain)
and for zt >1 (prior loss)
Then the individual’s maximization problem is
subject to the budget constraint in (15.14) and the dynamics for zt in (15.5)
Bellman equation
Define ρt J(Wt, zt) as the derived utility of wealth function. Then we can
write the Bellman equation as
Taking the first-order conditions of (15.15) w.r.t. Ct and St we get (15.16)
and (15.17)
From (15.16) and (15.17) we can derive the envelope condition (see
handout)
Substituting this into (15.16) we obtain the Euler equation
What are the economic interpretations of (15.18) and (15.19)?
Using (15.18) and (15.19) in (15.17) we get
or
Equilibrium
• In equilibrium we can replace the representative agent’s consumption Ct
with the aggregate level of consumption, .
• Then, under the assumption that aggregate consumption is lognormally
distributed, we can also solve for the risk-free interest rate:
Using (15.1 ) and (15.10) we can simplify (15.21) to
or
Deriving the Pt/Dt ratio
Numerical solution algorithm
• (15.24) and (15.25) need to be solved jointly. BHS use an iterative
numerical technique to solve for f ( ), by guessing at f 0( ), solving for zt+1
in (15.25), finding a new solution f 1( ), and using the following recursive
equation until the f i( )function converges to an equilibrium.
Selected parameter values
• gc = 1.84%; σc = 3.79%; γ = 1.0; ρ = 0.98 jointly imply that Rf -1
= 3.86% ( a low risk free rate)
• λ = 2.25 (indicates how keenly losses are felt relative to gains,
w/o prior losses or gains (Tversky & Kahneman, 1972)
• k = 3 (the penalty parameter to reflect how much more
painful losses are if they come after other losses) – chosen to
either:
1) get λ close to 2.25 or
2) to bring the equity risk premium close to its empirical (historical)
value.
• b0 = range (0.7 – 100) (determines the relative importance of
the prospect theory term in the investor’s preferences over
financial wealth fluctuations)
Summary of Results
• BHS find that Pt/Dt = ft(zt) is a decreasing function of zt. If there are prior
gains from investing in the risky asset (and zt is low), the investor becomes
less risk averse and bids up the price of the risky asset.
• Using the estimated f( ) function, we can simulate the unconditional
distribution of stock returns from the random sequence of εt. The
resulting volatility of stock prices can be substantially higher than that of
the aggregate consumption series.
• Then, due to loss aversion, the model generates an equity risk premium
that will be significant for relatively low levels of risk aversion, even if
stocks do not have a high correlation with consumption (recall that a low
covariance suggests a low equity risk premium in the standard
consumption asset pricing model) .
• The BHS model also implies predictability of stock returns - - returns are
higher following stock market crashes and smaller following expansions.
Therefore, stock returns are negatively correlated at long horizons.