investors. Then there exists a range of low aggregate dividend in which the
most risk averse investor buys a high share of the available claims so that the
aggregate RRA approaches her high level of constant RRA. This investor
dominates the market in this range. In the range of high dividends, this
investor buys a small share of the available claims so that her influence on
the market disappears. In this range the least risk averse investor dominates
the market. Hence there is a shift from a high risk aversion to a low risk
aversion regime. This shift may explain a crash.
The lemma does not prove that in some dividend range aggregate RRA
declines very rapidly. We therefore investigate this issue by simulating the
equilibrium allocation and the corresponding RRA.
4 Simulation
In this section we illustrate our results by some simulations. First, we derive
aggregate RRA assuming investors with constant RRA, but the level of RRA
varies across investors. Second, we discuss the procedure used for simulation.
Third, we present the simulation results.
4.1 Aggregate Relative Risk Aversion
The simulation approximates the valuation in an infinite horizon setting by
a finite horizon setting. The price of the market portfolio at the horizon is
approximated by a function of the aggregate dividend paid at the horizon.
Investors trade claims on the dividends paid until the horizon and claims
on the horizon market portfolio. Once the stochastic discount factor for
the horizon date is known, the stochastic discount factors for the preceding
dates can be derived from no-arbitrage. Therefore we need to derive the
stochastic discount factor for the horizon date or, equivalently, the aggregate
RRA for the horizon date. Assume that there are three investors with
different levels of constant relative risk aversion. (γ1 ,γ2,γ3) denotes the
vector of these levels. We derive the pareto-efficient allocation of claims
on the horizon market portfolio by using the social planner model. The
planner allocates these claims to the investors so as to maximize the weighted
sum of the investors’ utility subject to the constraint that the sum of all
claims equals the exogenous supply of claims. (1 /λi) is the state-independent
weight attached by the social planner to investor i. λi /λj can be interpreted
as the ratio of investor i’s over investor j’s expected marginal utility in
equilibrium. λi is higher, the smaller the wealth that investor i allocates to
claims on the horizon market portfolio. Since this wealth is determined by
the equilibrium allocation, this is also true of λi . Yet, the simulation takes
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