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Defining bad news: changes in return distributions that decrease risky asset demand.

Article Excerpt
1. Introduction

What changes in the distribution of risky asset returns cause investors to reduce their demand for risky assets? In addition to understanding individual portfolio choice, understanding such changes in distribution is also helpful in understanding the equilibrium price of the market portfolio. The aggregate quantity of the market portfolio of risky assets is fixed in the short run: changes that reduce the demand for the risky asset in the two-asset portfolio problem reduce the equilibrium price of the market portfolio.

We study a von Neumann-Morgenstern expected-utility-maximizing investor choosing an optimal portfolio consisting of one risky and one riskless asset. As a comparative statics exercise, we shift the rate of return distribution for the risky asset. What are the necessary and sufficient conditions on the shift in distribution for all strictly risk-averse investors to reduce their demand for the risky asset? When does such a reduction in demand make investors worse off?

Gollier (1995) provides a solution for the necessary and sufficient conditions to reduce demand in the form of integral conditions on the distribution functions before and after the shift. In contrast, our goal is to provide a random variable characterization of the necessary and sufficient conditions. To see the difference between the integral conditions given by Gollier (1995) and our random variable characterization, it is helpful to consider an analogy to the necessary and sufficient conditions for second-order stochastic dominance (SSD) in Rothschild and Stiglitz (1971).

One form of the necessary and sufficient conditions for SSD is expressed as a condition on the integrals of the cumulative distribution function. Such a condition is similar to the form of Gollier's (1995) condition. Rothschild and Stiglitz (1971) also show that a necessary and sufficient condition for SSD is that one random variable dominates another random variable by SSD if and only if the latter random variable is distributed as the former random variable plus white noise. This form of the condition for SSD is similar in spirit to the form of our random variable conditions for a decrease in demand for the risky asset. In other words, if SSD can be interpreted as adding a noise variable to the return, can decreased demand for the risky asset be characterized as adding some sort of random variable to the return? We provide such a characterization.

We report a random variable characterization of the necessary and sufficient shifts in distribution showing explicitly how to take the initial random variable for the excess rate of return on the risky asset and construct a new random variable for the excess rate of return on the risky asset, such that all risk-averse investors reduce risky asset demand. We also provide conditions for such a shift to decrease risky asset demand and to decrease the expected utility of all strictly risk-averse investors--our definition of bad news.

2. Literature Review

First- and second-order stochastic dominance reductions in the risky asset's rate of return distribution do not always reduce risky asset demand for all risk-averse investors. One approach to deriving comparative statics in the portfolio problem is to look for restrictions on the class of utility functions that lead to intuitive comparative static properties. For example, Rothschild and Stiglitz (1971), Kira and Ziemba (1980), and Hadar and Seo (1990) show that decreasing absolute risk aversion along with a coefficient of relative risk aversion bounded by one are sufficient conditions for a second-order shift to reduce demand, and Fishburn and Porter (1976) show that similar conditions are needed for a first-order shift to reduce demand.

Another approach to deriving comparative statics in the portfolio problem is to impose conditions on the shifts that result in reductions in demand for all risk-averse utility functions. Examples include the strong increase in risk introduced by Meyer and Ormiston (1985), simple increases in risk introduced by Dionne and Gollier (1996), relatively strong increases in risk introduced by Black and Bulkley (1989), and relatively weak increases in risk introduced Dionne et al. (1993). All of these shifts are sufficient to reduce demand for the risky asset in the two-asset portfolio problem with a riskless asset for all risk-averse investors.

Gollier (1995) provides necessary and sufficient conditions to reduce risky asset demand in the form of conditions on all the partial expectations of the distribution functions of returns before and after the shift. Gollier and Schlesinger (2002) use the conditions to characterize the shifts that reduce the risky asset price in an endowment economy. Athey (2002) provides general characterizations of the necessary and sufficient conditions for monotone comparative statics properties in stochastic optimization problems, and applies the characterizations to the portfolio problem relating the conditions to a single-crossing property on the change in distribution functions of the risky asset return.

Landsberger and Meilijson (1990) define a mean-preserving increase in risk around [upsilon]: a mean-preserving spread where probability mass is shifted around intervals whose closure contains [upsilon]. For [upsilon] = 0, the shifts satisfy the sufficient conditions to decrease risky asset demand for all risk-averse investors. We provide a random variable characterization of these shifts around [upsilon] = but allowing for the mean to decrease, and use the random variables as part of our construction of the random variables that characterize the necessary and sufficient conditions for all risk-averse investors to reduce risky asset demand. We also use techniques developed by Gollier and Kimball (1996) on the shifts to develop the necessary and sufficient conditions to reduce risky asset demand.

3. Assumptions and Notation

A strictly risk-averse investor maximizes the expected value of a strictly concave, increasing von Neumann-Morgenstern utility function U: [R.sup.++] [right arrow] R, assumed to be everywhere at least twice differentiable, with first derivative U': [R.sup.++] [right arrow] [R.sup.+]. The investor has initial wealth W, and allocates X to a risky asset with the remainder, W - X, allocated to a riskless asset. Our assumptions imply that the utility function has finite first derivative for all strictly positive consumption levels,

U' (C) 0, (1)

and allows for the possibility that the utility function satisfies the Inada condition that [lim.sub.[C[down arrow]0]]U'(C) = [infinity].

The risky asset has a rate of return given by the random variable [R.sup.y]. The riskless rate of return is, without loss of generality, set to zero: [R.sup.y] is therefore interpreted as an excess return. The excess rate of return on the risky asset [R.sup.y] has bounded support equal to [-1, +1], the cumulative probability distribution [F.sup.y](r) [equivalent to] Prob([R.sup.y] [less than or equal to] r) satisfies

< [F.sup.y] (0) < 1, (2)

and the bounded support assumption implies

< E[[R.sup.y]] < [infinity]. (3)

Condition (2) rules out arbitrage opportunities, and conditions (1) and (3) ensure that any risk-averse investor holds a positive amount of the risky asset in the optimal portfolio (Arrow 1971).

The investor's portfolio problem is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

Denoting the optimal level of risky investment by X*, the first-order condition, which is necessary and sufficient for a solution to (4), is

[[integral].sub.-1.sup.1] U' (W + rX*)r d[F.sup.y] (r) = 0. (5)

Suppose that the random variable describing the excess rate of return on the risky asset changes from [R.sup.y] to [R.sup.z], with the associated probability distribution changing from [F.sup.y] to [F.sup.z] By the strict concavity of U, the optimal holding of the risky asset weakly decreases if and only if the first-order condition in Equation (5) is less than or equal to zero when evaluated at X*,

[[integral].sub.-1.sup.1] U' (W + rX*) r...

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