V
This measure of risk aversion rises as the the surplus consumption ratio 5', declines, that is, as consumption declines toward habit.17
The requirement that consumption always be above habit is satisfied automatically in microeconomic models with exogenous asset returns and endogenous consumption, as in Constantinides (1990) and Sundarcsan (1989). It presents a more serious problem in models with exogenous consumption processes. To handle this problem Campbell and Cochrane (1995) specify a nonlinear process by which habit adjusts to consumption, remaining below consumption at all times. Campbell and Cochrane write down a process for the log surplus consumption ratio s, = log(.S',). They ¡fssumc that log consumption follows a random walk with drift g and innovation di+i, Ac,+i = g + . They propose an AR(1) model for s,:
ilere s is the steady-state surplus consumption ratio. The parameter 0 gov-
;''rns the persistence of the log surplus consumption ratio, while the sensi-ivity function X(s,) controls the sensitivity of s/+t and thus of log habit x,+i o innovations in consumption growth i/|+|.
Equation (8.4.16) specifics that today's habit is a complex nonlinear unction of current and past consumption. By taking a linear approxima-ion around the steady state, however, it may be shown-that (8.4.16) is ap-
,7Risk aversion may also he measured by the normalized cur vature of the value function maximized utility expressed as a ('unction of wealth), or by the volatility ol' the stochastic iscount factor, or by the maximum Sharpe ratio available tit asset markets. While these ijieasurcx of risk aversion are different from each other in this model, they all move inversely villi .Vf. Note that y, the curvature parameter in utility, is no longer a measure of risk aversion i l this model.
proximately a traditional habit-formation model in which loghabitresponds slowly to log consumption, x,+1 % [(l -4>)h -v tf] + <t>x, t (I - </.),, r\i
where k = lu( 1 — ,S') is the steady slate value of x-c. The problem with the traditional model (8.4.17) is that it allows consumption to lall below habit, resulting in infinite or negative marginal utility. A process for s, defined over the real line implies lhat consumption can never fall below habit.
Since habit is external, the marginal utility of consumption is u'(Q) = (C, — X,)~y = ,S', r C, Y. The stochastic discount factor is then
,, .M'(C,+ |) r. ( •">/+ I (-'<+ I \ V .
In the standard power utility model S, = 1, so the stochastic discount factor isjust consumption growth raised to the power-y. To get a volatile stochastic discount factor one needs a large value of y. In the habil-lbrmalion model one can instead get a volatile stochastic discount factor from a volatile surplus consumption ratio St.
The riskless real interest rale is related to the stochastic discount factor by (1 + K/+|) = 1 /(M,+ i). Taking logs, and using («.4.1(1) and (8.4.18), the log riskless real interest rate is r[+] = - log(S) + yg~ y(l - </>)(s, - s) - [*(.*,) 4- if • (8.4.19)
The first two terms on the right-band side of (8.4.19) are familiar from the power utility model (8.2.6), while the last two terms are new. The third term (linear in (s, — 1)) reflects intertemporal substitution, or mean-reversion in marginal utility. If the surplus consumption ratio is low, the marginal utility of consumption is high. However, the surplus consumption ratio is expectcd to revert to its mean, so marginal utility is expected to fall in the future. Therefore, the consumer would like to borrow and this drives up the equilibrium risk free interest rate. The fourth term (linear in iA(.5,)4-l]2) reflects precautionary savings. As uncertainty increases, consumers become more willing to save and this drives down the equilibrium riskless interest rate.
If this model is lo generate stable real interest rates like (hose observed in the data, the serial correlation parameter ip must be near one. Also, the sensitivity function A(s,) must decline with s, so that uncertainty is high when lilym ill lA/itiiil/iuuu Alltilels is tow ami the precautionary saving term offsets the intertemporal substitution let in. In fact, Campbell and Cochrane parametrize the A(.v,) function so that these two terms exactly offset each other everywhere, implying a constant risklcss interest rate.
Kvcu with a constant risklcss interest rate and random-walk consumption, (he external-habit model can produce a large equity premium, volatile stock prices, and predictable excess stock returns. The basic mechanism is time-variation in risk aversion. When consumption falls relative to habit, the resulting increase in risk aversion drives up the risk premium on risky assets such as stocks. This also drives down the prices of stocks, helping to explain why slock returns are so much more volatile than consumption growth or risklcss real interest rales.
Campbell and Cochrane (1995) calibrate their model to US data on consumption and dividends, solving for equilibrium stock prices in the tradition of Mehra and lYcscoit (1985). There is also some work on habit formation that uses actual stock return data in the tradition of Hansen and Singleton (198'>, 1983). Ileaton (1995), for example, estimates an internal-habit model allowing for lime-aggregation of the data and for some durability of those goods formally described as nondurable in the national income accounts. Durability can be thought of as the opposite of habit formation, in that consumption expenditure today lowers the marginal utility of consumption expenditure tomorrow. I leaton Itnds that durability predominates at high frequencies, and habit formation at lower frequencies. However his habit-formation model, like the simple power utility model, is rejected statistically.
Both these approaches assume that aggregate consumption is the driving process for marginal utility. An alternative view is that, for reasons discussed in Section K.'V'i, the consumption of stock market investors may not be adequately proxied by inacroecouomic data on aggregate consumption. Under this view the driving process for a habit-formation model should be a process with a reasonable mean and standard deviation, but need not be highly correlated with aggregate consumption.
S. I. 2 I \y< hologiitil Models of ¡'references
I'sychologisls and experimental economists have found that in experimental settings, people make choices that differ in several respects from the standard model «if expected utility. In response to these findings unorthodox "psychological" models of preferences have been suggested, and some recent research has begun to apply these models to asset pricing.1"
'"Useful general ii l. irni es include llitg.iiili anil Keller (1!>H7) and Kreps (I'.INH).
H. 4. More General Utility Functions
Psychological models may best be understood by comparing them to standard time-separable specification (8.1.1) in which an investor maximizes
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